We describe a short, direct, alternative to the DeTurck trick to prove the
uniqueness of solutions to a large class of curvature flows of all orders,
including the Ricci flow, the L^2 curvature flow, and other flows related
to the ambient obstruction tensor. Our approach is based on the analysis
of simple energy quantities defined in terms of the actual solutions to the
equations, and allows one to avoid the step -- itself potentially
nontrivial in the noncompact setting -- of solving an auxiliary parabolic
equation (e.g., a k-harmonic-map heat-type flow) in order to overcome the
gauge-invariance-based degeneracy of the original flow. We also
demonstrate that, by the consideration of a certain energy
quotient/frequency-type quantity, one can give a short and quantitative
proof (avoiding Carleman inequalities) of the global backward uniqueness of
solutions to a large class of these equations.