  Using the conjugation symmetry on Heegaard Floer complexes, we define a
three-manifold invariant called involutive Heegaard Floer homology, which is
meant to correspond to $\mathbb{Z}_4$-equivariant Seiberg-Witten Floer
homology. Further, we obtain two new invariants of homology cobordism,
$\underline{d}$ and $\bar{d}$, and two invariants of smooth knot concordance,
$\underline{V}_0$ and $\overline{V}_0$. We also develop a formula for the
involutive Heegaard Floer homology of large integral surgeries on knots. We
give explicit calculations in the case of L-space knots and thin knots. In
particular, we show that $\underline{V}_0$ detects the non-sliceness of the
figure-eight knot. Other applications include constraints on which large
surgeries on alternating knots can be homology cobordant to other large
surgeries on alternating knots.
