  We introduce an invariant of tangles in Khovanov homology by considering a
natural inverse system of Khovanov homology groups. As application, we derive
an invariant of strongly invertible knots; this invariant takes the form of a
graded vector space that vanishes if and only if the strongly invertible knot
is trivial. While closely tied to Khovanov homology -- and hence the Jones
polynomial -- we observe that this new invariant detects non-amphicheirality in
subtle cases where Khovanov homology fails to do so. In fact, we exhibit
examples of knots that are not distinguished by Khovanov homology but, owing to
the presence of a strong inversion, may be distinguished using our invariant.
This work suggests a strengthened relationship between Khovanov homology and
Heegaard Floer homology by way of two-fold branched covers that we formulate in
a series of conjectures.
