  A natural family of potentially 2-torsion elements in the integer homology
cobordism group consists of splices of knots with their mirrors. We show that
such 3-manifolds have locally trivial involutive Floer homology. We show some
related families of splices also have locally trivial involutive Floer
homology. Our arguments show that many gauge theoretic invariants also vanish
on these 3-manifolds.
