We present a comprehensive muon-spin-rotation/relaxation $(\ensuremath{\mu}\mathrm{SR})$ study of the high-temperature magnetic order ${(T}_{N}\ensuremath{\simeq}65\mathrm{K})$ in ${\mathrm{PrCu}}_{2},$ first observed quite unexpectedly in 1998 [A. Schenck et al., Phys. Rev. B $58,$ 5205 (1998)]. Measurements in zero and longitudinal fields as a function of temperature, orientation of the single-crystal sample, and strength of applied magnetic field ${H}_{\mathrm{ext}}$ confirmed the previous findings and allowed us to determine the properties of the internal spontaneous fields $\mathit{B}$ in detail. It is found that $\mathit{B}$ is strictly confined to the crystallographic $(a,c)$ plane and encloses on average an angle $\ensuremath{\theta}$ of $30\ifmmode^\circ\else\textdegree\fi{}\ensuremath{-}45\ifmmode^\circ\else\textdegree\fi{}$ with the a axis, depending on temperature. Both $|\mathit{B}|$ and $\ensuremath{\theta}$ display a large variation. This is consistent with, but does not unambiguously prove, the incommensurately modulated antiferromagnetic structure observed below $\ensuremath{\sim}50\mathrm{mK}$ by neutron scattering [S. Kawarazaki and J. Arthur, J. Phys. Soc. Jpn. $57,$ 1077 (1988)]. The estimated ordered electronic moment of $\ensuremath{\sim}0.29{\ensuremath{\mu}}_{B}$ is smaller than the value quoted by Kawarazaki et al. $(0.54{\ensuremath{\mu}}_{B}).$ The temperature dependence of $|\mathit{B}|$ is highly unusual and scales perfectly with the temperature dependence of the elastic constant ${C}_{66}$ [R. Settai et al., J. Phys. Soc. Jpn $67,$ 636 (1998)], implying that the ordered moment is proportional to the strain susceptibility involving the ${O}_{\mathrm{xy}}$ quadrupole moment. ${\ensuremath{\mu}}^{+}$ is also found to be exposed to fluctuating field components inducing spin-lattice relaxation. The relaxation rate $\ensuremath{\lambda}$ consists of a field-independent but rather anisotropic term and an essentially isotropic but field-dependent term. The first term reflects the Jahn-Teller transition at 7.5 K by a cusplike anomaly. Transverse-field measurements at 6 kOe and above 100 K allowed us to determine the anisotropic Knight shift, or the dipolar coupling tensor, from which it follows that ${\ensuremath{\mu}}^{+}$ are located at $4e$ sites. At lower temperatures, we find evidence that the contact-hyperfine coupling constant becomes temperature and orientation dependent.