  We examine the emergence of dynamical memory effects in quantum processes
having indefinite time direction and causal order. In particular, we focus on
the class of phase-covariant qubit channels, which encompasses some of the most
significant paradigmatic open quantum system models. In order to assess the
memory in the time evolution of the system, we utilize the trace distance and
the entanglement based measures of non-Markovianity. While the indefinite time
direction is obtained through the quantum time flip operation that realizes a
coherent superposition of forward and backward processes, the indefinite causal
order is achieved via the quantum switch map, which implements two quantum
processes in a coherent superposition of their two possible orders. Considering
various different families of phase-covariant qubit channels, we demonstrate
that, when implemented on memoryless quantum processes, both the quantum time
flip and the quantum switch operations can generate memory effects in the
dynamics according to the trace distance based measure under certain
conditions. On the other hand, with respect to the entanglement based measure,
we show that neither the quantum time flip nor the quantum switch could induce
dynamical memory for any of the considered phase-covariant channels.
