  Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define
a class of orthogonal polynomials on elliptic curves and establish a
corresponding Riemann-Hilbert framework. We then focus on the special case,
defined by a constant weight function, and use the Riemann-Hilbert problem to
derive recurrence relations and differential equations for the orthogonal
polynomials. We further show that the sub-class of even polynomials is
associated to the elliptic form of Painlev\'e VI, with the tau function given
by the Hankel determinant of even moments, up to a scaling factor. The first
iteration of these even polynomials relates to the special case of Painlev\'e
VI studied by Hitchin in relation to self-dual Einstein metrics.
