  We prove that certain polynomials previously introduced by the author can be
identified with tau functions of Painlev\'e VI, obtained from one of Picard's
algebraic solutions by acting with a four-dimensional lattice of B\"acklund
transformations. For particular lines in the lattice, this proves conjectures
of Bazhanov and Mangazeev. As applications, we describe the behaviour of the
corresponding solutions near the singular points of Painlev\'e VI, and obtain
several new properties of our polynomials.
