  We study the response of spin and charge order in single crystals of
La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ to uniaxial stress, through $^{139}$La
nuclear magnetic resonance (NMR) and $^{63}$Cu nuclear quadrupole resonance
(NQR), respectively. In unstressed La$_{1.875}$Ba$_{0.125}$CuO$_{4}$, the
low-temperature tetragonal structure onsets below $T_{\text{LTT}} = 57$K, while
the charge order and the spin order transition temperatures are $T_\text{CO} =
54$K and $T_\text{SO} = 37$K, respectively. We find that uniaxial stress along
the [110] lattice direction strongly suppresses $T_\text{CO}$ and
$T_{\text{SO}}$, but has little effect on $T_\text{LTT}$. In other words, under
stress along [110] a large splitting ($\approx 21$K) opens between
$T_\text{CO}$ and $T_{\text{LTT}}$, showing that these transitions are not
tightly linked. On the other hand, stress along [100] causes a slight
suppression of $T_\text{LTT}$ but has essentially no effect on $T_\text{CO}$
and $T_{\text{SO}}$. Magnetic field $H$ along [110] stabilizes the spin order:
the suppression of $T_\text{SO}$ under stress along [110] is slower under $H
\parallel [110]$ than $H \parallel [001]$. We develop a Landau free energy
model and interpret our findings as an interplay of symmetry breaking terms
driven by the orientation of spins.
