  Spontaneous order of layer pseudospins in two-dimensional bilayers is common
in quantum Hall systems, where it is responsible for hysteretic responses to
gate fields in states with Ising order, and giant drag voltages in states with
XY (spontaneous inter-layer phase coherence) order. In this article we predict
that layer pseudospin order will also occur in double-moir\'e strongly
correlated two-dimensional electron systems. We comment on similarities and
differences in the competition between the two types of order in quantum Hall
and double-moir\'e systems, and relate our findings to previous work on
Falicov-Kimball models of electronic ferroelectrics.
