  Highly symmetric networks can exhibit partly symmetry-broken states,
including clusters and chimera states, i.e., states of coexisting synchronized
and unsynchronized elements. We address the $\mathbb{S}_4$ permutation symmetry
of four globally coupled Stuart-Landau oscillators and uncover an
interconnected web of differently symmetric solutions. Among these are chaotic
$2\!-\!1\!-\!1$ minimal chimeras that arise from $2\!-\!1\!-\!1$ periodic
solutions in a period-doubling cascade, as well as fully asymmetric chaotic
states arising similarly from periodic $1\!-\!1\!-\!1\!-\!1$ solutions. A
backbone of equivariant pitchfork bifurcations mediate between the two
cascades, culminating in equivariant pitchforks of chaotic attractors.
