  We give an algebraic/geometric characterization of the classical
pseudodifferential operators on a smooth manifold in terms of the tangent
groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show
that a properly supported semiregular distribution on $M\times M$ is the
Schwartz kernel of a classical pseudodifferential operator if and only if it
extends to a smooth family of distributions on the range fibres of the tangent
groupoid which is homogeneous for the $\mathbb{R}^\times_+$-action modulo
smooth functions. Moreover, we show that the basic properties of
pseudodifferential operators can be proven directly from this characterization.
Finally, we show that with the appropriate generalization of the tangent
bundle, the same definition applies without change to define pseudodifferential
calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.
