  We establish $({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-dualities between
quantum Gaudin models with irregular singularities. Specifically, for any $M, N
\in {\mathbb Z}_{\geq 1}$ we consider two Gaudin models: the one associated
with the Lie algebra ${\mathfrak{gl}}_M$ which has a double pole at infinity
and $N$ poles, counting multiplicities, in the complex plane, and the same
model but with the roles of $M$ and $N$ interchanged. Both models can be
realized in terms of Weyl algebras, i.e., free bosons; we establish that, in
this realization, the algebras of integrals of motion of the two models
coincide. At the classical level we establish two further generalizations of
the duality. First, we show that there is also a duality for realizations in
terms of free fermions. Second, in the bosonic realization we consider the
classical cyclotomic Gaudin model associated with the Lie algebra
${\mathfrak{gl}}_M$ and its diagram automorphism, with a double pole at
infinity and $2N$ poles, counting multiplicities, in the complex plane. We
prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie
algebra ${\mathfrak{sp}}_{2N}$, with a double pole at infinity and $M$ simple
poles in the complex plane. In the special case $N=1$ we recover the well-known
self-duality in the Neumann model.
