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

Spectral Data of Special Orthogonal Higgs Bundles and Hecke Modification