  We derive several identities involving Ikeda and Naruse's $K$-theoretic Schur
$P$- and $Q$-functions. Our main result is a formula conjectured by Lewis and
the second author which expands each $K$-theoretic Schur $Q$-function in terms
of $K$-theoretic Schur $P$-functions. This formula extends to some more general
identities relating the skew and dual versions of both power series. We also
prove a shifted version of Yeliussizov's skew Cauchy identity for symmetric
Grothendieck polynomials. Finally, we discuss some conjectural formulas for the
dual $K$-theoretic Schur $P$- and $Q$-functions of Nakagawa and Naruse. We show
that one such formula would imply a basis property expected of the
$K$-theoretic Schur $Q$-functions.
