  We study the tuning effect of a transverse magnetic field on the confinement
of spinons in the infinite XXZ spin-1/2 chain. The spinon confinement in this
model takes place in the gapped antiferromagnetic phase upon application of a
staggered longitudinal magnetic field. The tuning transverse magnetic field has
mutually orthogonal uniform and staggered components. The energy spectra of the
two-spinon bound states (the `mesons') in the confinement regime are
analytically calculated in this model using two different perturbative schemes.
The first one applies in the extreme anisotropic (Ising) limit and employs the
inverse anisotropy constant as a small parameter. The second perturbative
scheme, which applies at any anisotropy in the gapped antiferromagnetic domain,
exploits the integrability of the XXZ spin chain at zero magnetic field. The
small parameters in the second technique are the components of the transverse,
and staggered longitudinal magnetic fields. It is shown, that the weak
transverse magnetic field mixes the transverse and longitudinal meson modes,
and leads to an avoided crossing of their energies upon increase of its
strength. The explicit formulas for the two-spinon contribution to the
dynamical structure factors of local spin operators are obtained as well in
this model in the weak confinement regime for wave-vectors close to the points
k = 0 and k = \pi.
