  Motivated by recent experiments reporting unconventional collective modes in
the charge density-wave (CDW) state of rare-earth tritellurides $R$Te$_3$, we
derive from a multi-orbital microscopic model on the square net a CDW
Ginzburg-Landau theory that allows for non-trivial orbital order. Our analysis
reveals unconventional CDWs where order parameters with distinct orbital
character coexist due to an approximate symmetry of the low-energy model, which
becomes exact in the limit of nearest-neighbor-only hopping and decoupled
$p_x$, $p_y$ orbitals. Because of this coexistence, the resulting CDW pattern
displays an orbital texture that generally breaks additional symmetries of the
lattice besides those explicitly broken by the CDW wave-vector. In particular,
we find two competing phases that differ in whether they break or preserve
inversion and vertical mirror symmetries. We explain the mechanisms that favor
each outcome, and discuss experimental probes that can distinguish the
different phases.
