  We study equivariant Seiberg-Witten Floer theory of rational homology
$3$-spheres in the special case where the group action is given by an
involution. The case of involutions deserves special attention because we can
couple the involution to the charge conjugation symmetry of Seiberg-Witten
theory. This leads to new Floer-theoretic invariants which we study and apply
in a variety of applications. In particular, we construct a series of
delta-invariants $\delta^E_*, \delta^R_*, \delta^S_*$ which are the equivariant
equivalents of the Ozsv\'ath-Szab\'o $d$-invariant. The delta-invariants come
in three types: equivariant, Real and spin depending on the type of the
spin$^c$-structure involved. The delta-invariants satisfy many useful
properties, including a Froyshov-type inequality for equivariant cobordisms. We
compute the delta-invariants in a wide range of examples including: equivariant
plumbings, branched double covers of knots and equivariant Dehn surgery. We
also consider various applications including obstructions to extending
involutions over bounding $4$-manifolds, non-smoothable involutions on
$4$-manifolds with boundary, equivariant embeddings of $3$-manifolds in
$4$-manifolds and non-orientable surfaces bounding knots.
