Suppose we have a square torus (i.e. complex structure $\tau=i$) which is fibred trivially over a circle of radius $4R$. We then use the $\mathbb{Z}_4$ rotational symmetry of the torus lattice to orbifold this bundle by a $\mathbb{Z}_4$ action as follows: given local coordinate $(y, z)$, where $y$ is the circle coordinate and $z$ is a complex coordinate of the torus, the identification is $(y,z) \sim (y+2\pi R, e^{i\pi/2}z)$, i.e. a quarter-shift in the base and a $\pi/2$ rotation in the torus.
Now, consider the same torus, but this time fibred non-trivially over a circle of radius $R$ with a $\mathbb{Z}_4$ twist, i.e. there is a monodromy when we go around the base circle of $(y, z) \sim (y+2\pi R, e^{i\pi/2}z)$.
Looking at the coordinate identifications, these two look like they are describing the same thing, just obtained in two slightly different ways. Is this obviously true, or am I missing something, and is it enough for the coordinate identifications to be the same for the two bundles to be identical?