I've touched on most of this before and anyone with any minimal familiarity with Marxist / post-Marxist thought will already get all of this - still, the topic has been subjected to such persistent and mendacious misrepresentation by liberal capitalists that it's worth reiterating, once again, what a left position on identitarian politics actually looks like. There is of course a spectrum of opinion here, but instead of attempting to be comprehensive, I'll try to be specific by speaking for myself.
Identitarianism primarily investigates relationships between identity, privilege, and oppression. It makes the utterly uncontroversial observation that some identities are subject to the domination of others, considers the intersectional relationships between different identities, and develops a politics of emancipation with these considerations in mind. Historically, identitarianism has primarily focused on issues of race, sex, class, and sexual orientation, though this scope is continually expanding to contemplate other dimensions of identity as well.
Personally, I'm in general agreement with liberals on a broad range of identitarian concerns. I recognize all of the usual categories of oppression (racism, sexism, homophobia, transphobia, ableism, nationalism, ageism, etcetera). These forces emerge from all kinds of fairly well understood factors ranging from historical bias to cognitive biases, and they structure our society into the familiar hierarchies that place (for example) straight white men at the top while subordinating everyone else. They play out in almost every aspect of our society: in the way that we understand each other, talk to each other, portray each other in the media, build relationships with each other, employ each other, vote for each other, in the way that we structure our institutions and policies, and so on.
Accordingly, an identitarian politics has to take all of this into account, grapple with, and ameliorate all of the different ways that oppression emerges in our society. To take a simple example, identitarianism has to acknowledge that histories of genocide, slavery and racism in our society have entrenched all kinds of systematic and cultural injustices against people of color that demand a dedicated political response. Though there's considerable controversy (even among liberals) about the appropriate approach, my sense is that reparations have to be a major part of this effort, along with all kinds of direct interventions into policing, employment, housing, education, and so on.
This, again, is all completely orthodox and mainstream liberal identitarianism. The positions and proposals I have in mind here map directly onto the standard platforms of any Democratic politicians and liberal activists you can name.
Liberal vs. left identitarianism
The major difference between liberal and left identitarianism, in my view, is that leftists have remained faithful to the identitarian tradition as expressed historically and in the overwhelming body of literature. Principally, this means that leftists have maintained their committment understanding power and oppression thoroughly by including class identity in their analysis, as all of the great identitarian scholars have always done - whereas liberals, by definition, neglect it.
The liberal, of course, will insist that they don't actually neglect class - which is true if by "class" we just mean "wealth". But here, I'm using class in the Marxist sense that means something like "one's identity as an owner or non-owner of the means of production".
This, revisionism aside, was always a major category of analysis in identitarian thought. Crucially, there is no identitarian reason to exclude it. Identitarianism doesn't impose some rule about what categories of oppression "count" and which ones we should ignore - one of its major strengths has always been its insistance on an expansive and inclusive conception of power that takes into consideration as many aspects of oppression as possible. If the proletariat have no place in identitarianism, it is not because of something about identitarianism, but because the proletariat does not in fact exist - because, that is, Marx's theory of class identity is wrong.
This point is crucial, because it marks what is (at least for me) the major distinction between liberal and left identitarians. Liberals definitionally reject the Marxist claim that ownership of the means of production plays a significant role in power and oppression. At the very most, they are willing to acknowledge a role for wealth in their identitarian analysis, distinguishing between (say) a rich disabled WOC and a poor one. But because they are capitalists, they cannot seriously consider the role that proletarian vs. bourgeois identity plays in power and oppression. Contrary to liberal portrayals of left thought as relatively narrow and exclusive, it is the liberal's theory of identitarianism that excludes a form of identity - class - that left identitarianism always includes.
Once one accepts a role for the proletariat in identitarian thought, the analysis and practice of identitarianism proceeds as usual. One conducts an intersectional investigation of how various forms of identity relate to each other, how their effects shape, overlap and amplify each other, and so on. From there, one develops a strategy for ending all of these various forms of oppression as thoroughly and expediently as possible, bearing in mind all the obvious considerations of severity, urgency, policy constraints, political obstacles, justice, etcetera.
Distinctively, one of the forms of identity the left incorporates into this intersectional analysis is class. The leftist investigates the way that class shapes identity and multiplies power and oppression, just as it would investigate the way that gender does this, or race, or anything else. That is how intersectionality works. And on that basis, it then develops a praxis for emancipation.
We know that race intersects with other forms of identity in all kinds of unique ways, and understanding them can give us unique and important insights into how oppression works. For instance, while men are generally privileged compared to women, race complicates that picture, and it becomes clear that a white woman may have privileges that a black man does not. The same kinds of insights also emerge when we investigate the role of class. Generally, being a member of the bourgeoisie (which controls the means of production and and can use that to exploit workers) is a position of privilege over being a member of the proletariat (which can only sell its labor as is always at the mercy of the bougeoisie). Again, an intersectional analysis of identity can complicate this picture. A member of the bourgeoisie may not have to work for a living, but he is more likely to get assaulted by cops he is black than if he's white, and more likely to be raped if she's a woman than if he's a man. A working class white woman is obviously in little danger of falling victim to racism, but she faces all kinds of serious disadvantages that Carlos Slim Helu does not.
All of these different dimensions of identity relate to each other in extraordinarily complicated ways. None of them completely account for or can be reduced to the effects of the others: economic forces do not entirely explain the problems of sexism, for example, and neither can sexism explain all of the problems of class. An intersectional analysis has to explore all of the historical, cultural, material, technological, biological, and psychological factors that go into the construction of these identities in order to develop a big picture about how they relate to each other and how we are to proceed in our politics.
Thus far, all of this has really just been an overcomplicated way of saying "we have to understand all of the complications of our world in order to proceed in a just and moral way". You can see this basic principle evolve all the way from its ancient philosophical origins into the identitarian / intersectional language that we now use to talk about politics. In common, both liberalism and leftism emerged during this evolution from phases of gross reduction and oversimplification. Even liberalism passed through a phase of an overly mechanistic and myopic focus on deterministic economism that persists in part to this very day: thus, we see the Freakonomics-style analyses of capitalism that continue to reduce problems of gender income inequality (for example) to the simple mechanics of supply and demand.
Leftism, for its part, also briefly passed through such a phase more than a century ago, arguing that class struggle within the material economy could account for all expressions of power and oppression. This perspective, as I've written before, has been dead for ages. Marxists like Gramsci incorporated matters of culture, race and ethnicity into his intersectional analysis of power; writers from the Frankfurt School added all kinds of considerations of sexuality and gender, among other things; and in the mid-to-late twentieth century, writers like Foucault elaborated and complicated left identitarianism so thoroughly that modern leftists debate the role of class in so-called post-Marxist thought almost constantly.
The left identitarian agenda
This debate is so extensive and complicated, encompassing a whole genre of modern left literature, that it would be pointless to try to capture or fairly represent it here: so again, I'll try to stick to my perspective. Today, I see the identitarian left agenda as encompassing three major planks:
1. The implementation of measures against oppression that happen to be compatible with liberalism (but do not belong to it), such as criminal justice reform, education campaigns, gun control, bathroom access laws, and so on. All of this is fairly uncontroversial, at least generally, though liberals tend to maintain (counterfactually) that the left opposes such measures.
2. Opposition to liberal and reactionary measures that would ultimately increase oppression. A basic example of this would be left identitarian efforts to defend and restore the Voting Rights Act. A more complicated example would be left identitarian opposition to NAFTA, which dramatically increased the class domination of workers throughout North America, which exasperated problems of racism against Hispanics, and which which had all kinds of other foreseeable consequences for the oppressed. Because liberals omit Marxist class identity from their analysis, they generally supported NAFTA on capitalist grounds, and did not anticipate all of its intersectional effects.
3. The implementation of emancipatory measures that are at odds with liberalism. This includes all kinds of things that would fight the problems of racism, sexism, homophobia, transphobia, ableism, and just about any other ism you can name - but that liberal identitarianism routinely rejects, since these measures would also fight classism. For example, the implementation of universal health care would disproportionately benefit women, people of color, the disabled, and all kinds of other oppressed identity groups, and significantly at the expense of able-bodied white men - but since universal health care is at odds with the privilege of the bourgeoisie, liberalism tends to resist it. More ambitious policies include things ranging from dramatically expanded affirmative action requirements to the abolition of private property - all measures that would erode all kinds of privilege that liberals nominally oppose, but that liberals reject since it would also erode the prrivilege of the bourgeoisie.
That's it. That's the left identitarian agenda, widely supported by women, people of color, the poor, the LGBT community, the stateless, the disabled, and so on, all over the world. This agenda is mostly opposed by a relatively small faction of people who, whatever their other identities, reject identitarian politics through their omission of Marxist class as a form of identity. Casually, we can describe these (from a global perspective, wealthy) folks as liberal identitarians, though they've fundamantally rejected identitarian thought; accordingly, we could just call them liberals, though this doesn't quite capture their politics either. They're fundamental alignment with the interests and privilege of the bourgeoisie is what distinguishes them from true identitarians, and for that reason the most precise word for them is "capitalists".