  We study a system of purely repulsive spherical self-propelled particles in
the minimal set-up inducing Motility-Induced Phase Separation (MIPS). We show
that, even if explicit alignment interactions are absent, a growing order in
the velocities of the clustered particles accompanies MIPS. Particles arrange
into aligned or vortex-like domains. Their sizes increase as the persistence of
the self-propulsion grows, an effect that is quantified studying the spatial
correlation function of the velocities. We explain the velocity-alignment by
unveiling a hidden alignment interaction of the Vicsek-like form, induced by
the interplay between steric interactions and self-propulsion. As a
consequence, we argue that the MIPS transition cannot be fully understood in
terms of a scalar field, the density, since the collective orientation of the
velocities should be included in effective coarse-grained descriptions.
