  We prove that the Lipshitz-Ozsv\'ath-Thurston correspondence between extended
type D structures of knot complements and $\mathbb{F}[U, V]/(UV)$ knot Floer
complexes can be arranged so that $\iota_K$-invariant splittings of knot Floer
chain complexes correspond to $\iota_{S^3 \setminus K}$-invariant splittings of
bordered Floer homology of knot complements. For patterns satisfying the
satellite extension property, which include cabling patterns, this provides a
novel way to compute the involutive knot Floer homology of satellites from that
of their companions. As a topological application, we show that our results can
be applied to construct infinitely many examples of exotic pairs of
contractible 4-manifolds which remain exotic after one stabilization. Along the
way, we also establish first order naturality of bordered Floer homology.
