  We study the scaling properties of the differential cross section of elastic
proton-proton ($pp$) and proton-antiproton ($p\bar p$) collisions at high
energies. We introduce a new scaling function, that scales -- within the
experimental errors -- all the ISR data on elastic $pp$ scattering from
$\sqrt{s} = 23.5$ to $62.5$ GeV to the same universal curve. We explore the
scaling properties of the differential cross-sections of the elastic $pp$ and
$p\bar p$ collisions in a limited TeV energy range. Rescaling the TOTEM $pp$
data from $\sqrt{s} = 7$ TeV to $2.76$ and $1.96$ TeV, and comparing it to D0
$p\bar p$ data at $1.96$ TeV, our results provide an evidence for a $t$-channel
Odderon exchange at TeV energies, with a significance of at least 6.26$\sigma$.
We complete this work with a model-dependent evaluation of the domain of
validity of the new scaling and its violations. We find that the $H(x)$ scaling
is valid, model dependently, within $200$ GeV $ \leq \sqrt{s} \leq$ $ 8$ TeV,
with a $-t$ range gradually narrowing with decreasing colliding energies.
