  The microscopic modeling of spin-orbit entangled $j=1/2$ Mott insulators such
as the layered hexagonal Iridates Na$_2$IrO$_3$ and Li$_2$IrO$_3$ has spurred
an interest in the physics of Heisenberg-Kitaev models. Here we explore the
effect of lattice distortions on the formation of the collective spin-orbital
states which include not only conventionally ordered phases but also gapped and
gapless spin-orbital liquids. In particular, we demonstrate that in the
presence of spatial anisotropies of the exchange couplings conventionally
ordered states are formed through an order-by-disorder selection which is not
only sensitive to the type of exchange anisotropy but also to the relative
strength of the Heisenberg and Kitaev couplings. The spin-orbital liquid phases
of the Kitaev limit -- a gapless phase in the vicinity of spatially isotropic
couplings and a gapped Z$_2$ phase for a dominant spatial anisotropy of the
exchange couplings -- show vastly different sensitivities to the inclusion of a
Heisenberg exchange. While the gapless phase is remarkably stable, the gapped
Z$_2$ phase quickly breaks down in what might be a rather unconventional phase
transition driven by the simultaneous condensation of its elementary
excitations.
