It's done often by those of all beliefs. Actually, by trying to paint this as something one side does, in a way you are doing the very same thing.
Conservatives don't do these things, though.
Conservative Right and Conservative Left can easily compromise on abortion, for example. The finished product being responsible yet unlimited use pre-viability, with reasonable restrictions to protect both the mother's health and the potential life inside her post-viability.
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Liberal Right want's marriage to be damn-near if not exclusively religious, with of course all the legal benefits. Meanwhile, Liberal Left would like to completely dissolve marriage in every way if they can't get whatever token minority group they're pandering for the votes of, access to 'marriage'.
Conservatives, however, pretty much agree that variety is fine so long as the emphasis of marriage remains the raising and socializing of children. Other than that, alternative couplings are tolerable even if personal religious preferences differ.
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Conservatives of every party, even Conservative Loosertarians, tend to get along with each-other quite well. A Liberal Looserterian wants all drugs legalized across the board, even-through this means no one would ever again need a prescription. Liberal Right want to keep beating their heads against the wall with the failed "war on drugs" and put you in prison for 12 years if you have so much as 1 pot seed.
Conservative Right and Conservative Loosertarians have common ground in legalizing recreational drugs like pot, while placing more emphasis on rehabilitation over prison time for users of hard drugs like meth.
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Conservatives play nice with other Conservatives, but liberals fight to the death with Liberals of other tribe-er, I mean parties.