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.
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