  We investigate interaction-driven integer quantum Hall states realized in
Landau levels of monolayer graphene when two out of its four nearly degenerate
spin-valley flavors are filled. By employing a model that accounts for
interactions beyond pure delta-functions as well as Zeeman and
substrate-induced valley potentials, we demonstrate the existence of a delicate
competition of several phases with spontaneous generation of spin-valley
entanglement, akin to the spontaneous appearance of spin-orbit coupling driven
by interactions. We encounter a particular phase that we term the
entangled-Kekul\'{e}-antiferromagnet (E-KD-AF) which only becomes spin-valley
entangled under the simultaneous presence of Zeeman and substrate potentials,
because it gains energy by simultaneously canting in the spin and valley
spaces, by combining features of a canted anti-ferromagnet and a canted
Kekul\'{e} state. We quantify the degree of spin-valley entanglement of the
many competing phases by computing their bipartite concurrence.
