  We give a new, elementary proof that Khovanov homology with
$\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This
proof also gives a strategy to prove Baldwin and Levine's conjecture that
$\delta$--graded knot Floer homology is mutation--invariant. Using the Clifford
module structure on $\widetilde{\text{HFK}}$ induced by basepoint maps, we
carry out this strategy for mutations on a large class of tangles. Let $L'$ be
a link obtained from $L$ by mutating the tangle $T$. Suppose some rational
closure of $T$ corresponding to the mutation is the unlink on any number of
components. Then $L$ and $L'$ have isomorphic $\delta$--graded
$\widehat{\text{HFK}}$-groups over $\mathbb{Z}/2\mathbb{Z}$ as well as
isomorphic Khovanov homology over $\mathbb{Q}$. We apply these results to
establish mutation--invariance for the infinite families of Kinoshita-Terasaka
and Conway knots. Finally, we give sufficient conditions for a general
Khovanov-Floer theory to be mutation--invariant.
