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