  In the Hamburg cold atom experiment with orbital states in an optical
lattice, $s$- and $p$-orbital atomic states hybridize between neighbouring
sites. In this work we show how this alternation of sites hosting $s$- and
$p$-orbital states gives rise to a plethora of different magnetic phases,
quantum and classical. We focus on phases whose properties derive from
frustration originating from a competition between nearest and next nearest
neighbouring exchange interactions. The physics of the Mott insulating phase
with unit filling is described by an effective spin-1/2 Hamiltonian showing
great similarities with the $J_1$-$J_2$ model. Based on the knowledge of the
$J_1$-$J_2$ model, together with numerical simulations, we discuss the
possibility of realising a quantum spin liquid phase in the present optical
lattice system. In the superfluid regime we consider the parameter regime where
the $s$-orbital states can be adiabatically eliminated to give an effective
model for the $p$-orbital atoms. At the mean-field level we derive a
generalized classical $XY$ model, and show that it may support maximum
frustration. When quantum fluctuations can be disregarded, the ground state is
expected to be a spin glass. Even with quantum fluctuations present it has been
debated whether a spin liquid may persist at the point of full frustration.
