  We establish Lagrangian formulae for energy conservation anomalies involving
the discrepancy between short-time two-particle dispersion forward and backward
in time. These results are facilitated by a rigorous version of the
Ott-Mann-Gaw\c{e}dzki relation, sometimes described as a "Lagrangian analogue
of the 4/5ths law". In particular, we prove that for any space-time $L^3$ weak
solution of the Euler equations, the Lagrangian forward/backward dispersion
measure matches on to the energy defect in the sense of distributions. For
strong limits of $d\geq3$ dimensional Navier-Stokes solutions the defect
distribution coincides with the viscous dissipation anomaly. The Lagrangian
formula shows that particles released into a $3d$ turbulent flow will initially
disperse faster backward-in-time than forward, in agreement with recent
theoretical predictions of Jucha et. al (2014). In two dimensions, we consider
strong limits of solutions of the forced Euler equations with increasingly
high-wavenumber forcing as a model of an ideal inverse cascade regime. We show
that the same Lagrangian dispersion measure matches onto the anomalous input
from the infinite-frequency force. As forcing typically acts as an energy
source, this leads to the prediction that particles in $2d$ typically disperse
faster forward in time than backward, which is opposite to what occurs in $3d$.
Time-asymmetry of the Lagrangian dispersion is thereby closely tied to the
direction of the turbulent cascade, downscale in $d\geq 3$ and upscale in
$d=2$. These conclusions lend support to the conjecture of Eyink & Drivas
(2015) that a similar connection holds for time-asymmetry of Richardson
two-particle dispersion and cascade direction, albeit at longer times.
