  By a bifurcation argument we prove that the capillary-gravity Whitham
equation features asymmetrical periodic travelling wave solution of arbitrarily
small amplitude. Such waves exist only in the weak surface tension regime
$0<T<\frac{1}{3}$ and are necessarily bimodal; they are located at double
bifurcation points satisfying a certain symmetry breaking condition. Our
bifurcation argument is an extension of the one applied by Ehrnstr\"om et al to
find symmetric waves: Here, two additional scalar equations arise. Combining
the variational structure of our problem with its translation symmetry, we show
that these two additional equations are in fact linearly dependent, and can (at
'symmetry breaking' bifurcation points) be solved by incorporating the surface
tension as a bifurcation parameter. Contrary to the symmetric case, only very
specific modal pairs $(k_1,k_2)$ give rise to (small) asymmetrical periodic
waves and we here provide a partial characterization of such pairs.
