Recently, the n\(^3\)He collaboration reported a measurement of the parity-violating (PV) proton directional asymmetry \(A_{\mathrm {PV}} = (1.55\pm 0.97~\mathrm {(st\ at)} \pm 0.24~\mathrm {(sys)})\times 10^{-8}\) in the capture reaction of \({}^3\)He\((\vec {n},{\mathrm p}){}^3\)H at meV incident neutron energies. The result increased the limited inventory of precisely measured and calculable PV observables in few-body systems required to further understand the structure of hadronic weak interaction. In this letter, we report the experimental and theoretical investigation of a parity conserving (PC) asymmetry \(A_{\mathrm {PC}}\) in the same reaction (the first ever measured PC observable at meV neutron energies). As a result of S- and P-wave mixing in the reaction, the \(A_{\mathrm {PC}}\) is inversely proportional to the neutron wavelength \(\lambda\). The experimental value is \((\lambda\times A_{\mathrm {PC}})\equiv\beta= (-1.97 \pm 0.28~\mathrm{(stat)}\pm 0.12~\mathrm{(sys)}) \times 10^{-6}\) Amstrongs. We present results for a theoretical analysis of this reaction by solving the four-body scattering problem within the hyperspherical harmonic method. We find that in the \({}^3\)He\((\vec {n},{\mathrm p}){}^3\)H reaction, \(A_{\mathrm {PC}}\) depends critically on the energy and width of the close \(0^-\) resonant state of \({}^4\)He, resulting in a large sensitivity to the spin-orbit components of the nucleon-nucleon force and even to the three-nucleon force. The analysis of the accurately measured \(A_{\mathrm {PC}}\) and \(A_{\mathrm {PV}}\) using the same few-body theoretical models gives essential information needed to interpret the PV asymmetry in the \({}^3\)He\((\vec {n}, {\mathrm p}){}^3\)H reaction.