  Active matter has been the object of huge amount of research in recent years
for its important fundamental and applicative properties. In this paper we
investigate active suspensions of micro-swimmers through direct numerical
simulation, so that no approximation is made at the continuous level other than
the numerical one. We consider both pusher and puller organisms, with a
spherical or ellipsoidal shape. We analyse the velocity and the characteristic
scales for an homogeneous two-dimensional suspension and the effective
viscosity under shear. We bring evidences that the complex features displayed
are related to a spontaneous breaking of the time-reversal symmetry. We show
that chaos is not a key ingredient, whereas a large enough number of
interacting particles and a non-spherical shape are needed to break the
symmetry and are therefore at the basis of the phenomenology. Our numerical
study also shows that pullers display some collective motion, though with
different characteristics from pushers.
