  We revisit the connection between relativistic orbital precession, the
Laplace-Runge-Lenz symmetry, and the $t$-channel discontinuity of scattering
amplitudes. Applying this to scalar-tensor theories of gravity, we compute the
conservative potential and orbital precession induced by both
conformal/disformal-type couplings at second Post-Minkowskian order
($\mathcal{O} \left( G_N^2 \right)$), complementing the known third/first order
Post-Newtonian results. There is a particular tuning of the conformal coupling
for which the precession vanishes at leading PN order, and we show that this
coincides with the emergence of a Laplace-Runge-Lenz symmetry and a
corresponding soft behaviour of the amplitude. While a single scalar field
inevitably breaks this symmetry at higher PN orders, certain supersymmetric
extensions have recently been shown to have an exact Laplace-Runge-Lenz
symmetry and therefore classical orbits do not precess at any PN order. This
symmetry can be used to relate scattering amplitudes at different loop orders,
and we show how this may be used to bootstrap the (classically relevant part of
the) three-loop $2 \to 2$ scattering of charged black holes in $\mathcal{N}=8$
supergravity from existing two-loop calculations.
