My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for your time and help! Here goes:

Consider the confluent hypergeometric equation

$zy''+(c-z)y'-ay=0$.

This has solutions of the form

$C_1 \Phi (a,c;z) + C_2 z^{1-c} \Phi (1+a-c,2-c;z)$,

where

$\Phi(a,c;z) = \sum_{k=0}^{\infty} \frac{(a)_k}{(c)_k} \frac{z^k}{k!}, \quad(a)_k = a(a+1)(a+2) \cdots (a+k-1)$

It is also an example of a Fuchsian differential equation.

My question is about how far the evident relation between the two linearly independent solutions to the confluent hypergeometric equation generalizes to other Fuchsian differential equations. One can think of the confluent hypergeometric equation

$zy''+ ((2-c)-z)y'- (1+a-c)y=0$

as dual in a sense to the first equation I listed, because the $\Psi$ solutions to the first equation are related to the $\Phi$ solutions of the second, and vice versa.

Because of Fuchs's Theorem guaranteeing (generically) two solutions to a Fuchsian differential equation with different asymptotic behavior for small values of the independent variable, it seems to me likely that this must be an example of some more general "duality" for a larger class of Fuchsian differential equations. Is it true that for some class of Fuchsian differential equations, one kind of solution can be regarded as "primary" (analogous to $\Phi$), and the other (the "secondary" $\Psi$ one) can be gotten in terms of the "primary" solution to another differential equation dual to the first, like for the confluent hypergeometric case?

I have a suspicion that monodromy groups might accomplish something like this, but I just have a very vague inkling of what these might be and no idea where to start reading. If someone could confirm or deny my suspicion and maybe point me to a reference, I would be very grateful!

Thanks again, and I apologize for the long post!