  Inspired by the notions of local equivalence in monopole and Heegaard Floer
homology, we introduce a version of local equivalence that combines odd
Khovanov homology with equivariant even Khovanov homology into an algebraic
package called a local even-odd (LEO) triple. We get a homomorphism from the
smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$
of such triples. We give several versions of the $s$-invariant that descend to
$C_{LEO}$, including one that completely determines whether the image of a knot
$K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the
power of these invariants in obstructing sliceness, both statistically and for
some interesting knots studied by Manolescu-Piccirillo. Along the way, we
explore several variants of this local equivalence group, including one that is
totally ordered.
