New polymeric liquid crystals can be treated as standard nematic liquid crystals when only their bulk properties are at issue, but they exhibit peculiar surface properties. The most striking one is that biaxial distributions may be induced on a confining surface. On continuously varying the surface anchoring conditions, we find a first-order phase transition from planar to homeotropic alignment in the bulk. Moreover, the decay towards these uniaxial states is radically different in the two cases: it is asymptotically exponential in the former, whereas it happens abruptly at a finite depth in the latter. There is precisely one surface biaxial distribution that induces bistability between these decay modes: it depends on the elastic constants in the Landau-de Gennes free energy functional. The analysis of the model we propose can prove useful in detecting the sign of the difference between splay and bend constants.