  Just as a symmetric surface with separating fixed locus halves into two
oriented bordered surfaces, an arbitrary symmetric surface halves into two
oriented symmetric half-surfaces, i.e. surfaces with crosscaps. Motivated in
part by the string theory view of real Gromov-Witten invariants, we previously
introduced moduli spaces of maps from surfaces with crosscaps, developed the
relevant Fredholm theory, and resolved the orientability problem in this
setting. In this paper, we determine the relative signs of the automorphisms of
these moduli spaces induced by interchanges of boundary components of the
domain and by the anti-symplectic involution on the target manifold, without
any global assumptions on the latter. As immediate applications, we describe
sufficient conditions for the moduli spaces of real genus 1 maps and for real
maps with separating fixed locus to be orientable; we treat the general genus
2+ case in a separate paper. Our sign computations also lead to an extension of
recent Floer-theoretic applications of anti-symplectic involutions and to a
related reformulation of these results in a more natural way.
