on the horns of a dilemma

:  one of the usually paired bony processes that arise from the head of many ungulates and that are found in some extinct mammals and reptiles: such as

:  one of the permanent paired hollow sheaths of keratin usually present in both sexes of cattle and their relatives that function chiefly for defense and arise from a bony core anchored to the skull — see cow illustration

:  antler

:  an argument presenting two or more equally conclusive alternatives against an opponent

:  a usually undesirable or unpleasant choice

:  a situation involving such a choice