  We give a complete, self-contained computation of the spectral data
parametrising Higgs bundles in the generic fibres of the
$\mathrm{SO}_{2n+1}$-Hitchin fibration where the Higgs fields are $L$-twisted
endomorphisms. Although the spectral data is known in the literature, we
develop a new approach to spectral data, which takes advantage of Hecke
modification. Further, we present the computation for the $\mathrm{Sp}_{2n}$
and $\mathrm{SO}_{2n}$ cases while clarifying some aspects of the
correspondence which are not well explained in the pre-existing literature. We
also compute the number of connected components of the generic fibres, and
demonstrate Langlands duality in the fibres via the canonical duality in the
fibres.
