Graphs whose complement and square are isomorphic

We study square-complementary graphs, that is, graphs whose complement and square are isomorphic. We prove several necessary conditions for a graph to be square-complementary, describe ways of building new square-complementary graphs from existing ones, construct infinite families of square-complementary graphs, and characterize square-complementary graphs within various graph classes. The bipartite case turns out to be of particular interest. We also exhibit a square-complementary graph with a triangle, thus answering in the affirmative a question asked independently by Baltić et al. and by Capobianco and Kim.