We consider solitons in a nonlinear optical fiber with a single polarization in a region of parameters where it carries exactly two distinct modes, viz., the fundamental one and the first-order helical mode. From the viewpoint of applications to dense-WDM communication systems, this opens a way to double the number of channels carried by the fiber. Aside from that, experimental observation of helical (spinning) solitons (that can be launched and detected, using helicity-generating phase masks) and collisions between them and with fundamental solitons in (ordinary or hollow) optical fibers is an issue of fundamental interest, especially because it has been very recently found that spatiotemporal spinning solitons in bulk optical media with various nonlinearities are unstable. We introduce a system of coupled nonlinear Schrödinger equations for fundamental and helical modes, computing nonstandard values of the cross-phase-modulation coupling constants in it, and investigate, analytically and numerically, results of "complete" and "incomplete" collisions between solitons carried by the two modes. We conclude that the collision-induced crosstalk is partly attenuated in comparison with the usual WDM system, which sometimes may be crucially important, preventing merger of the colliding solitons into a breather. The interaction between the two modes is found to be additionally strongly suppressed in comparison with that in the WDM system in the case when a dispersion-shifted or dispersion-compensated fiber is used.