  Low-energy elastic and inelastic scattering in the Ps(1$s$)-Ps(2$s$) channel
is treated in a four-body hyperspherical coordinate calculation. Adiabatic
potentials are calculated for triplet-triplet, singlet-singlet, and
singlet-triplet spin symmetries in the spin representation of coupled electrons
and coupled positrons, with total angular momentum $L=0$ and parity equal to
$+1$. The s-wave scattering lengths for the asymptotic Ps(1$s$)-Ps(2$s$)
channel are calculated for each spin configuration. Results obtained for the
s-wave scattering lengths are $a_{\mathrm{TT}}=$~$7.3(2)a_0-i0.02(1)a_0$,
$a_{\mathrm{SS}}=$~$13.2(2)a_0-i0.9(2)a_0$, and $a_{\mathrm{ST}}=$~$9.7(2)a_0$
for each spin configuration. Spin recoupling is implemented to extract the
scattering lengths for collisions of Ps in different spin configurations
through properly symmetrized unitary transformations. Calculations of
experimentally relevant scattering lengths and cross-sections are carried-out
for Ps atoms initially prepared in different uncoupled spin states.
