In the framework of compressional magnetohydrodynamics (MHD), we numerically studied the commonly accepted presumption that the Alfvénic turbulence is generated by the collisions between counter-propagating Alfvén waves (AWs). In the conditions typical for the low-beta solar corona and inner solar wind, we launched two counter-propagating AWs in the three-dimensional simulation box and analyzed polarization and spectral properties of perturbations generated before and after AW collisions. The observed post-collisional perturbations have different polarizations and smaller cross-field scales than the original waves, which supports theoretical scenarios with direct turbulent cascades. However, contrary to theoretical expectations, the spectral transport is strongly suppressed at the scales satisfying the classic critical balance of incompressional MHD. Instead, a modified critical balance can be established by colliding AWs with significantly shorter perpendicular scales. We discuss consequences of these effects for the turbulence dynamics and turbulent heating of compressional plasmas. In particular, solar coronal loops can be heated by the strong turbulent cascade if the characteristic widths of the loop substructures are more than ten times smaller than the loop width. The revealed new properties of AW collisions have to be incorporated in the theoretical models of AW turbulence and related applications.