  We study the entanglement of families of Unruh modes in the Bell states
$|\Phi^\pm\rangle =1/\sqrt{2}(|00\rangle\pm|11\rangle)$ and
$|\Psi^\pm\rangle=1/\sqrt{2}(|01\rangle\pm|10\rangle)$ shared by two
accelerated observers and find fundamental differences in the robustness of
entanglement against acceleration for these states. States $\Psi^\pm$ are
entangled for all finite accelerations, whereas, due to the Unruh effect,
states $\Phi^\pm$ lose their entanglement for finite accelerations. This is
true for Bell states of two bosonic modes, as well as for Bell states of a
bosonic and a fermionic mode. Furthermore, there are also differences in the
degradation of entanglement for Bell states of fermionic modes. We reveal the
origin of these distinct characteristics of entanglement degradation and
discuss the role that is played by particle statistics. Our studies suggest
that the behavior of entanglement in accelerated frames strongly depends on the
occupation patterns of the constituent states, whose superposition constitutes
the entangled state, where especially states $\Phi^\pm$ and $\Psi^\pm$ exhibit
distinct characteristics regarding entanglement degradation. Finally, we point
out possible implications of hovering over a black hole for these states.
