  Polyhomogeneous symbols, defined by Kohn-Nirenberg and H\"ormander in the
60's, play a central role in the symbolic calculus of most pseudodifferential
calculi. We prove a simple characterisation of polyhomogeneous functions which
avoids the use of asymptotic expansions. Specifically, if $U$ is open subset of
$\mathbb{R}^d$, then a polyhomogeneous symbol on $U \times \mathbb{R}^d$ is
precisely the restriction to $t=1$ of a function on $U \times \mathbb{R}^{d+1}$
which is homogeneous for the dilations of $\mathbb{R}^{d+1}$ modulo Schwartz
class functions. This result holds for arbitrary graded dilations on the vector
space $\mathbb{R}^d$. As an application, using the generalisation of A.~Connes'
tangent groupoid for a filtered manifold, we show that the Heisenberg calculus
of Beals and Greiner on a contact manifold or a codimension 1 foliation
coincides with the groupoid calculus of Van Erp and the second author.
