Suppose that E and E' denote real Banach spaces with dimension at least 2 and that D {subset of with not equal to} E and D' {subset of with not equal to} E' are uniform domains with homogeneously dense boundaries. We consider the class of all ϕ-FQC (freely ϕ-quasiconformal) maps of D onto D' with bilipschitz boundary values. We show that the maps of this class are η-quasisymmetric. As an application, we show that if D is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class ϕ-FQC by the smaller class of M-QH maps, we show that M-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if f is a ϕ-FQC map which maps D onto itself with identity boundary values, then there is a constant C, depending only on the function ϕ, such that for all x ∈ D, the quasihyperbolic distance satisfies k(x, f(x)) ≤ C.