  With much higher sensitivities due to coherence effects, it is often assumed
that the first evidence for direct dark matter detection will come from
experiments probing spin-independent interactions. We explore models that would
be invisible in such experiments, but detectable via spin-dependent
interactions. The existence of much larger (or even only) spin-dependent
tree-level interactions is not sufficient, due to potential spin-independent
subdominant or loop-induced interactions. We find that in such a way most
models with detectable spin-dependent interactions would also generate
detectable spin-independent interactions. Models in which a light pseudoscalar
acts as the mediator seem to uniquely evade this conclusion. We present a
particular viable dark matter model generating such an interaction.
