Using first-principles calculations, we analyze the structural properties of tungsten trioxide WO$_{3}$. Our calculations rely on density functional theory and the use of the B1-WC hybrid functional, which provides very good agreement with experimental data. We show that the hypothetical high-symmetry cubic reference structure combines several ferroelectric and antiferrodistortive (antipolar cation motions, rotations, and tilts of oxygen octahedra) structural instabilities. Although the ferroelectric instability is the largest, the instability related to antipolar W motions combines with those associated to oxygen rotations and tilts to produce the biggest energy reduction, yielding a $\textit{P}$2$_{1}$/$\textit{c}$ ground state. This nonpolar $\textit{P}$2$_{1}$/$\textit{c}$ phase is only different from the experimentally reported $\textit{Pc}$ ground state by the absence of a very tiny additional ferroelectric distortion. The calculations performed on a stoichiometric compound so suggest that the low-temperature phase of WO$_{3}$ is not intrinsically ferroelectric and that the experimentally observed ferroelectric character might arise from extrinsic defects such as oxygen vacancies. Independently, we also identify never observed $\textit{R3m}$ and $\textit{R3c}$ ferroelectric metastable phases with large polarizations and low energies close to the $\textit{P}$2$_{1}$/$\textit{c}$ ground state, which makes WO$_{3}$ $\textbf{a}$ potential antiferroelectric material. The relative stability of various phases is discussed in terms of the anharmonic couplings between different structural distortions, highlighting a very complex interplay. ; Other ; The work was supported by the ARC project AIMED and the F.R.S-FNRS PDR projects HiT4FiT and MaRePeThe. Calculations have been performed within the PRACE project TheDeNoMo and relied on the Céci facilities funded by F.R.S-FNRS (Grant No. 2.5020.1) and Tier-1 supercomputer of the Fédération Wallonie-Bruxelles funded by the Walloon Region (Grant No. 1117545). E.K.H.S. is grateful for support to EPSRC and the Leverhulme trust. H.H. thanks the AVERROES-ERASMUS Mundus project.