  We formulate a correspondence between SU(2) monopole chains and ``spectral
data'', consisting of curves in $\mathbb{CP}^1\times\mathbb{CP}^1$ equipped
with parabolic line bundles. This is the analogue for monopole chains of
Donaldson's association of monopoles with rational maps. The construction is
based on the Nahm transform, which relates monopole chains to Higgs bundles on
the cylinder. As an application, we classify charge $k$ monopole chains which
are invariant under actions of $\mathbb{Z}_{2k}$. We present images of these
symmetric monopole chains that were constructed using a numerical Nahm
transform.
