Both the intermediate-coupling theory and a strong-coupling theory appropriate for a moving polaron give an anomalous energy-momentum relation, $E(P)$, for the piezoelectric polaron. This relation starts out quadratic at small $P$, but at large $P$ it asymptotes to a straight line with slope equal to the speed of sound $s$. It is shown that the states represented by this anomalous $E$ vs. $P$ relation are such that the electron is clothed by its concomitant lattice distortion and that this lattice distortion becomes larger as the polaron velocity $v\ensuremath{\rightarrow}s$. This indicates that in this limit an increasing number of phonons are gathering in the "phonon cloud" around the electron. In both theories the constant lattice-potential surfaces become flattened in the forward and backward directions as the polaron starts to move.