  A series of Pr(TM)$_2$X$_{20}$ (with TM=Ti,V,Rh,Ir and X=Al,Zn) Kondo
materials, containing non-Kramers Pr$^{3+}$ $4f^2$ moments on a diamond
lattice, have been shown to exhibit intertwined orders such as quadrupolar
order and superconductivity. Motivated by these experiments, we propose and
study a Landau theory of multipolar order to capture the phase diagram and its
field dependence. In zero magnetic field, we show that different quadrupolar
states, or the coexistence of quadrupolar and octupolar orderings, may lead to
ground states with multiple broken symmetries. Upon heating, such states may
undergo two-step thermal transitions into the symmetric paramagnetic phase,
with partial restoration of broken symmetries in the intervening phase. For
nonzero magnetic field, we show the evolution of these thermal phase
transitions strongly depends on the field direction, due to clock anisotropy
terms in the free energy. Our findings shed substantial light on experimental
results in the Pr(TM)$_2$Al$_{20}$ materials. We propose further experimental
tests to distinguish purely quadrupolar orders from coexisting
quadrupolar-octupolar orders.
