  We attack the classification problem of multi-faced independences, the first
non-trivial example being Voiculescu's bi-freeness. While the present paper
does not achieve a complete classification, it formalizes the idea of lifting
an operator on a pre-Hilbert space in a "universal" way to a larger product
space, which is key for the construction of (old and new) examples. It will be
shown how universal lifts can be used to construct very well-behaved
(multi-faced) independences in general. Furthermore, we entirely classify
universal lifts to the tensor product and to the free product of pre-Hilbert
spaces. Our work brings to light surprising new examples of 2-faced
independences. Most noteworthy, for many known 2-faced independences, we find
that they admit continuous deformations within the class of 2-faced
independences, showing in particular that, in contrast with the single faced
case, this class is infinite (and even uncountable).
