  We present a study of the resonant structure of the decay $D^0 \to K_{\rm
L}^0\pi^+\pi^-$, using quantum-correlated $D^0\bar{D}^0$ data produced at
$\sqrt{s}=3.773$ GeV. The data sample was collected by the BESIII experiment
and corresponds to an integrated luminosity of $2.93$ fb$^{-1}$. This study is
the first amplitude analysis of a decay mode involving a $K_{\rm L}^0$, which
also results in the first measurement of the complex $\textit{U-spin breaking
parameters}$ ($\hat{\rho}$) related to various $\mathit{CP}$-eigenstate
resonant modes through which the three-body decay proceeds. The moduli of the
$\hat{\rho}$ parameters have central values in a wide range from $0.4$ to
$12.1$, which indicates substantial U-spin symmetry breaking. We present the
fractional resonant contributions and average strong-phase parameters over
regions of phase space for both $K_{\rm S}^0\pi^+\pi^-$ and $K_{\rm
L}^0\pi^+\pi^-$ modes. We also report the ratio of the branching fractions
between $K_{\rm L}^0\pi^+\pi^-$ and $K_{\rm S}^0\pi^+\pi^-$ decay modes and the
$\mathit{CP}$-even fraction of the $K_{\rm L}^0\pi^+\pi^-$ state calculated
using the U-spin breaking parameters.
