have it both ways
oil and water
irreconcilable
incompatible
two sides of the same coin
lucus a non lucendo
contraspective
out of both sides of one's mouth
contradictionary
contrasted
Antarctic
nonidentity
dilemma
incompatibly
on the horns of a dilemma
sugar and spice
orthogonal
vagrate
double
contradictional
contrary
dissemblance
oxymoron
tromboon
conflict
nondifference
multiproblematic
never the twain shall meet
betweenity
pushmi-pullyu
discrepant
between the devil and the deep blue sea
misrelate
contradictively
contradictio in terminis
bejuggle
intercontradictory
repugnance
gainsaying
match made in hell
Tweedledum and Tweedledee
butter one's bread on both sides
undermatch
ill-matched
disparate
contradictorial
counterdistinguish
inconsistent
conceptual inverse
contradictory