  We study two different actions on the moduli spaces of logarithmic
connections over smooth complex projective curves. Firstly, we establish a
dictionary between logarithmic orbifold connections and parabolic logarithmic
connections over the quotient curve. Secondly, we prove that fixed points on
the moduli space of connections under the action of finite order line bundles
are exactly the push-forward of logarithmic connections on a certain unramified
Galois cover of the base curve. In the coprime case, this action of finite
order line bundles on the moduli space is cohomologically trivial.
