[en] We demonstrate numerically and experimentally an efficient broadband asymmetric transmission (AT) by bilayer cross-shaped metamaterials for linearly polarized plane waves, attributed to a huge conversion capability between the cross polarizations. Physically, the phenomenon of AT results from the symmetry-broken geometry in this metamaterial. Due to the interaction of long and short-arm resonances of the twisted crosses, the AT bandwidth can be up to $33\mathrm{\%<!--\; \%\; -->}$ of the central wavelength, corresponding to a maximum cross-polarization conversion rate of $60\mathrm{\%<!--\; \%\; -->}$. In particular, the broad bandwidth can be manipulated by the short-arm length, which produces an opportunity in designing a polarization-based spectral filter or other microwave devices. (paper)