A spherical graph is a graph in which every interval is antipodal. Spherical graphs are an interesting generalization of hypercubes (a graph G is a hypercube if and only if G is spherical and bipartite). Besides hypercubes, there are many interesting examples of spherical graphs that appear in design theory, coding theory and geometry e.g., the Johnson graphs, the Gewirtz graph, the coset graph of the binary Golay code, the Gosset graph, and the Schläfli graph, to name a few. In this paper we study the structure of spherical graphs. In particular, we classify a subclass of these graphs consisting of what we call the strongly spherical graphs. This allows us to prove that if G is a triangle-free spherical graph then any interval in G must induce a hypercube, thus providing a proof for a conjecture due to Berrachedi, Havel and Mulder.