  Reformulation of conventional beam definitions into their bidirectional
versions and use of Hertz potentials make beam fields exact vector solutions to
Maxwell's equations. This procedure is applied to higher-order elegant
Laguerre-Gaussian beams of TM and TE polarization. Their vortex and anti-vortex
co-axial compositions of equal and opposite topological charges are given in a
closed analytic form. Polarization components of the composed beams are
specified by their radial and azimuthal indices. The longitudinal components
are common for beam compositions of both types, meanwhile their transverse
components are different and comprise two - nonparaxial and paraxial - separate
parts distinguished by a paraxial parameter and its inverse, respectively. The
new solutions may appear useful in modelling and tailoring of arbitrary vector
beams.
