Data-driven reduced-order models often struggle with high-dimensional nonlinear systems sensitive to low-variance coordinates, which are typically truncated. To address this, we use ideas from balanced truncation and active subspaces to identify low-dimensional coordinate systems that balance adjoint-based sensitivity information with state variance along trajectories. Our method, analogous to balanced truncation, replaces system Gramians with state and adjoint-based gradient covariance matrices, maintaining key transformation laws. We also present a further refinement whereby the resulting oblique projection is iteratively optimized to minimize forecasting error. We demonstrate and compare these techniques with other methods on a challenging toy problem and a nonlinear axisymmetric jet flow simulation with 100,000 state variables.