Motivated by recent progress in studying the duality-symmetric models of nonlinear electrodynamics, we revert to the auxiliary tensorial (bispinor) field formulation of the $O(2)$ duality proposed by us in Ivanov and Zupnik [Nucl. Phys. B618, 3 (2001); Yad. Fiz. 67, 2212 (2004)]. In this approach, the entire information about the given duality-symmetric system is encoded in the $O(2)$-invariant interaction Lagrangian which is a function of the auxiliary fields ${V}_{\ensuremath{\alpha}\ensuremath{\beta}}$, ${\overline{V}}_{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\alpha}}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\beta}}}$. We extend this setting to duality-symmetric systems with higher derivatives and show that the recently employed ``nonlinear twisted self-duality constraints'' amount to the equations of motion for the auxiliary tensorial fields in our approach. Some other related issues are briefly discussed and a few instructive examples are explicitly worked out.