  The purpose of this paper is to study motivic aspects of the Hitchin system
for $\mathrm{GL}_n$. Our results include the following. (a) We prove the
motivic decomposition conjecture of Corti-Hanamura for the Hitchin system; in
particular, the decomposition theorem associated with the Hitchin system is
induced by algebraic cycles. This yields an unconditional construction of the
motivic perverse filtration for the Hitchin system, which lifts the
cohomological/sheaf-theoretic perverse filtration. (b) We prove that the
inverse of the relative Hard Lefschetz symmetry is induced by a relative
algebraic correspondence, confirming the relative Lefschetz standard conjecture
for the Hitchin system. (c) We show a strong perversity bound for the
normalized Chern classes of a universal bundle with respect to the motivic
perverse filtration; this specializes to the sheaf-theoretic result obtained
earlier by Maulik-Shen. (d) We prove a $\chi$-independence result for the
relative Chow motives associated with Hitchin systems.
  Our methods combine Fourier transforms for compactified Jacobian fibrations
associated with integral locally planar curves, nearby and vanishing cycle
techniques, and a Springer-theoretic interpretation of parabolic Hitchin moduli
spaces.
