  We uncover new behaviors of the transmission of information by three quantum
channels in superposition of causal orders subject to some level of noise. We
find that the transmission can exhibit three different behaviors as the level
of noise is varied. This multifold behavior can be explained by the different
equivalence classes of quantum switch matrices related to specific combinations
of causal orders. We classify these matrices using their characteristic
polynomials and matrix invariants, and we calculate analytical expressions for
the Holevo information in three representative cases. Our results are a step
forward to understand and harness quantum control of causal orders with
different levels of noise. We also study the Holevo information as function of
a continuous order parameter and analyse transitions at integer values.
