If your compact Riemann surfaces $M$ and $N$ have a hyperbolic metric in which the boundary curves are totally geodesic of the same length, then this follows from the Fenchel-Nielsen coordinate parametrization of Teichmuller space. Any compact Riemann surface with boundary has a uniformization to a hyperbolic surface with totally geodesic boundary (one may see this by doubling, and applying the uniformization theorem). If you perform this uniformization, and then compare the two sides, your identification between boundary components may differ from the parametrization given by Fenchel-Nielsen coordinates (for example, the lengths of the boundary components may differ), and so it's more complicated to see how the conformal structure changes. In fact, if $M$ or $N$ is a disk or annulus, then the conformal structure may not be changing at all. If $M$ and $N$ both have Euler characteristic less than zero, then large twists should change the conformal structure. For example, a twist by $2\pi$ will change the Riemann surface by Dehn twist, and therefore gives a different conformal structure (up to isotopy).
There's also a slight issue of moduli, in that a rotation by $2\pi$ changes the point in Teichmuller space, but not in moduli space. I'm implicitly assuming you're asking for the Teichmuller parameter. If you're asking for the parameter in moduli space, then the twist will take you along a non-trivial closed loop in moduli space, so at least there will be uncountably many different conformal moduli as you twist.
If your compact Riemann surfaces $M$ and $N$ have a hyperbolic metric in which the boundary curves are totally geodesic of the same length, then this follows from the Fenchel-Nielsen coordinate parametrization of Teichmuller space. Any compact Riemann surface with boundary has a uniformization to a hyperbolic surface with totally geodesic boundary (one may see this by doubling, and applying the uniformization theorem). If you perform this uniformization, and then compare the two sides, your identification between boundary components may differ from the parametrization given by Fenchel-Nielsen coordinates (for example, the lengths of the boundary components may differ), and so it's more complicated to see how the conformal structure changes. In fact, if $M$ or $N$ is a disk or annulus, then the conformal structure may not be changing at all. If $M$ and $N$ both have Euler characteristic less than zero, then large twists should change the conformal structure. For example, a twist by $2\pi$ will change the Riemann surface by Dehn twist, and therefore gives a different conformal structure (up to isotopy).
There's also a slight issue of moduli, in that a rotation by $2\pi$ changes the point in Teichmuller space, but not in moduli space. I'm implicitly assuming you're asking for the Teichmuller parameter.