  We introduce quasi-BPS categories for twisted Higgs bundles, which are
building blocks of the derived category of coherent sheaves on the moduli stack
of semistable twisted Higgs bundles over a smooth projective curve. Under some
condition (called BPS condition), the quasi-BPS categories are non-commutative
analogues of Hitchin integrable systems.
  We begin the study of these quasi-BPS categories by focusing on a conjectural
symmetry which swaps the Euler characteristic and the weight. Our conjecture is
inspired by the Dolbeault Geometric Langlands equivalence of Donagi--Pantev, by
the Hausel--Thaddeus mirror symmetry, and by the $\chi$-independence phenomenon
for BPS invariants of curves on Calabi-Yau threefolds.
  We prove our conjecture in the case of rank two and genus zero. In higher
genus, we prove a derived equivalence of rank two stable twisted Higgs moduli
spaces as a special case of our conjecture.
  In a separate paper, we prove a version of our conjecture for the topological
K-theory of quasi-BPS categories and we discuss the relation between quasi-BPS
categories and BPS invariants of the corresponding local Calabi-Yau threefold.
