We study the structure of the hydrogen atom when placed in a high-frequency, superintense laser field, within the framework of a nonperturbative theory recently developed for this purpose. The theory predicts that in the high-frequency limit the atom is stable against decay by multiphoton ionization, and that its structure is determined by a time-independent Schr\"odinger equation containing a ``dressed'' Coulomb potential. The laser frequency \ensuremath{\omega} and the intensity I enter only combined in the parameter ${\ensuremath{\alpha}}_{0}$=${I}^{1/2}$${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}2}$ a.u. We first analyze the symmetry of the eigenvalue problem for the case of linear polarization under consideration and adopt an appropriate classification scheme for the levels. The small-${\ensuremath{\alpha}}_{0}$ limit of the levels is obtained analytically. In the large-${\ensuremath{\alpha}}_{0}$ limit scaling laws are derived for the ${\ensuremath{\alpha}}_{0}$ dependence of the eigenvalues and eigenfunctions. At finite ${\ensuremath{\alpha}}_{0}$ we have carried out a very accurate numerical computation over an extended range of ${\ensuremath{\alpha}}_{0}$ values (0\ensuremath{\le}${\ensuremath{\alpha}}_{0}$\ensuremath{\le}200 a.u.) for a number of symmetry manifolds, by diagonalization of the Hamiltonian in a Gaussian basis. The correlation diagrams relating the small- and large-${\ensuremath{\alpha}}_{0}$ limits exhibit several avoided crossings. The binding energies show an overall decrease with ${\ensuremath{\alpha}}_{0}$, in some cases preceded by an increase through a maximum. For the ground state this decrease is quite steep. The extreme distortion of the atomic structure accompanying it is studied. It is shown that, with increasing ${\ensuremath{\alpha}}_{0}$, the (oscillating) electronic cloud undergoes radiative stretching, which eventually culminates at large ${\ensuremath{\alpha}}_{0}$ in its splitting into two parts (dichotomy). The consequences of our findings for the experimental energy spectrum of the ejected electrons are considered.