  We show that all the common definitions of quasiregular mappings $f\colon
M\to N$ between two equiregular subRiemannian manifolds of homogeneous
dimension $Q\geq 2$ are quantitatively equivalent with precise dependences of
the quasiregularity constants. As an immediate consequence, we obtain that if
$f$ is $1$-quasiregular according to one of the definitions, then it is also
$1$-quasiregular according to any other definition. In particular, this
recovers a recent theorem of Capogna et al. on the equivalence of
$1$-quasiconformal mappings.
  Our main results answer affirmatively a few open questions from the recent
research. The main new ingredient in our proofs is the distortion estimates for
particular local extensions of the horizontal metrics. These extensions are
named "Popp extensions", and based on these extensions, we introduce a new
natural and invariant definition of quasiregularity in the equiregular
subRiemannian setting. The analysis on Popp extensions and on the implied
distortion is also of independent interest.
