The class CFL\cap co-CFL Is anything nontrivial known about the class $\mathrm{CFL}\cap \mathrm{coCFL}$? In particular, is it known whether $\mathrm{CFL}\cap \mathrm{coCFL} = \mathrm{DCFL}$ (certainly the reverse containment is obvious.)? I hope I'm not being stupid here--don't laugh at me too much if this is totally trivial. I ask this because I'm trying to pin down a commutative closure of a regular language over a binary alphabet, and it's clear that it lies inside this set, via a result showing that slip-languages over an alphabet of size two have context-free commutative closure.
For every finite, non-unary alphabet, the language of all palindromes is not in DCFL, but in the intersection of coCFL and CFL.