So much to chew over here.
There’s a running (accurate) joke on the Right (the folks at National Review like to comment on this)-- the Left only likes conservatives after they’re dead. Eisenhower, Goldwater, Reagan, Buckley… many on the Left hated all of them to varying degrees, but the moment they’re no longer around to defend themselves (or advocate for conservative causes), they become useful as a silent weapon against current conservative thinkers and politicians.
It’s lazy, and inaccurate.
As long as we’re on typology, for American conservatism in general, it is often argued on the Right that they are the “true” liberals. Full disclosure: I’m conservative, so perhaps I’m biased, but from a historical perspective this is accurate. Liberalism used to be just that, a solitary word without adjectives, to distinguish itself from European monarchism and early radicalism (anarchism & Jacobinism). This was the American political tradition, informed by the likes of Locke and Burke. It wasn’t until the introduction of Socialist and Communist movements in the late 19th century that the Left found itself in a typological schism of how to define themselves, and contrast themselves with the other great reform movement, i.e. liberalism. The American and British non-radical Left adopted “social liberalism” and “progressivism” as their banners, but for much of the late 19th and early 20th centuries, many on the Left swam in a muddied ideological sea. By the 20th century, the Left had metamorphosed into a wide variety of political thought: communism, socialism, fascism, social liberalism, progressivism (with the members of the Left moving across those categories quite freely, back and forth, at different points in their lives). In America, social liberalism eventually prevailed, while in other nations other Leftist movements achieved dominance (communism in the Soviet Union, fascism in Italy and Germany, socialism in France & to a lesser extent the UK).
By this point, there were two types of liberals in the U.S.: “classical” liberals and “social” liberals. However, since classical liberalism was the pre-existing political philosophy, by definition, they became “conservatives.” Eventually, social liberals just became liberals. And that’s how you ended up here today.
Of course, it’s frustrating that the simple labels are used in a vacuum when in fact it’s important to understand their origins and contexts. An American Republican and a Taliban leader are both “conservative” insofar as they act to conserve their own political traditions. The obvious difference, of course, is that both are conserving radically different things. That doesn’t stop the angry Left from using “conservative” as a shorthand for “all things we hate,” with the aforementioned example being prevalent (“the Taliban wing of the Republican party”… oh, right, I forgot about them… WTF?!).
Anyway, back to the OP… I’m not sure there’s been much of a rapprochement / reassessment of conservatives by liberals or vice versa. What’s happened instead has either been the fading of memory-- people forget why they ever argued in the first place over what appear in hindsight to be petty disputes-- or the aforementioned disingenuous “dead conservatives/liberals are good conservatives/liberals” argument.
Personally, I miss the Harry Truman / JFK / Scoop Jackson movement in the Democratic Party. Members of the early 20th century Democratic Party flirted heavily with socialism, communism and fascism, indulging in many varieties of “grass is greener” envy. It wasn’t until the Cold War began that the Democratic Party consciously chose to separate and distinguish itself from the international Left (which had come to be dominated by Soviet communism). That movement survived only for a short time-- 1946 through roughly 1972, with some lights lasting through the 1980s-- but I believe that it’s no coincidence that an era where the politics “stopped at the water’s edge” coincided with the greatest proportional rise of American power and influence.
Admittedly, that’s a bit like stealing bases-- “I respect the liberals who agreed with conservatives!”-- but hey, my opinion.
Unfortunately, today, there’s no real hope of finding that kind of consensus again. 9/11 might have offered that opportunity, but it was spoiled. One can quibble over where the blame belongs, but the facts remain that the American Left is far more comfortable looking overseas for its liberal inspirations than it is here at home (i.e., Europe has universal health care, Europe hates the war in Iraq, China can order people to build solar panels, etc.).