The description of the 𝑇⁢𝑇 deformation in terms of two-dimensional gravity is analyzed from the Hamiltonian point of view, in a manner analogous to the Arnowitt-Deser-Misner description of general relativity. We find that the Hamiltonian constraints of the theory imply relations between target-space momentum at finite volume that are equivalent to the 𝑇⁢𝑇 finite-volume flow equations. This fully quantum 𝑇⁢𝑇 result emerges already at the classical level within the gravitational theory. We exemplify the analysis for the case when the undeformed sector is a collection of 𝐷 −2 free massless scalars, where it is shown that—somewhat nontrivially—the target-space two-dimensional Poincaré symmetry is extended to 𝐷 dimensions. The connection between canonical quantization of this constrained Hamiltonian system and previous path integral quantizations is also discussed. We extend our analysis to the “gravitational” description of 𝐽⁢𝑇 -type deformations, where it is found that the flow equations obtained involve deformations that twist the spatial boundary conditions.