  Observations of diffuse radio emission in galaxy clusters indicate that
cosmic-ray electrons are accelerated on $\sim$Mpc scales. However, protons
appear to be accelerated less efficiently since their associated hadronic
$\gamma$-ray emission has not yet been detected. Inspired by recent
particle-in-cell simulations, we study the cosmic-ray production and its
signatures under the hypothesis that the efficiency of shock acceleration
depends on the Mach number and on the shock obliquity. For this purpose, we
combine ENZO cosmological magneto-hydrodynamical simulations with a Lagrangian
tracer code to follow the properties of the cosmic rays. Our simulations
suggest that the distribution of obliquities in galaxy clusters is random to
first order. Quasi-perpendicular shocks are able to accelerate cosmic-ray
electrons to the energies needed to produce observable radio emission. However,
the $\gamma$-ray emission is lowered by a factor of a few, $\sim$3, if
cosmic-ray protons are only accelerated by quasi-parallel shocks, reducing (yet
not entirely solving) the tension with the non-detection of hadronic
$\gamma$-ray emission by the \textit{Fermi}-satellite.
