The paper investigates the existence of interfacial (Stoneley-type) acoustic waves localised at the internal boundary between two semi-infinite superlattices which are adjoined with each other to form one-dimensional phononic bicrystal. Each superlattice is a periodic sequence of perfectly bonded homogeneous and/or functionally graded layers of general anisotropy. Given any asymmetric arrangement of unit cells (periods) of superlattices, it is found that the maximum number of interfacial waves, which can emerge at a fixed tangential wavenumber for the frequency varying within a stopband, is three for the lowest stopband and six for any upper stopband. Moreover, we show that this number of three or six waves in the lowest or upper stopband, is actually the maximum for the number of waves occurring per stopband in a given bicrystal plus their number in the “complementary” bicrystal, which is obtained by swapping upper and lower superlattices of the initial one (so that both bicrystals have the same band structure). An example is provided demonstrating attainability of this upper bound, i.e. the existence of six interfacial waves in a stopband. The results obtained under no assumptions regarding the material anisotropy are also specified to the case of monoclinic symmetry leading to acoustic mode decoupling.Read less <