Pronounced anisotropy of magnetic properties and complex magnetic order of a new oxi-halide compound Co-7(TeO3)(4)Br-6 has been investigated by powder and single-crystal neutron-diffraction, magnetization, and ac susceptibility techniques. Anisotropy of susceptibility extends far into the paramagnetic temperature range. A principal source of anisotropy are anisotropic properties of the involved octahedrally coordinated single Co2+ ions, as confirmed by angular-overlap-model calculations presented in this work. Incommensurate antiferromagnetic order sets in at T-N=34 K. The propagation vector is strongly temperature dependent reaching k(1)=[0.9458(6),0,0.6026(5)] at 30 K. A transition to a ferrimagnetic structure with k(2)=0 takes place at T-C=27 K. The magnetically ordered phase is characterized by very unusual anisotropy as well: while M-H scans along the b axis reveal spectacularly rectangular but otherwise standard ferromagnetic hysteresis loops, M-H studies along other two principal axes are perfectly reversible, revealing very sharp spin-flop (or spin-flip) transitions, such as in a standard antiferromagnet (or metamagnet). Altogether, the observed magnetic phenomenology is interpreted as an evidence of competing magnetic interactions permeating the system, in particular of the single-ion anisotropy energy and the exchange interactions. Different coordinations of the Co2+ ions involved in the low-symmetry C2/c structure of Co-7(TeO3)(4)Br-6 render the exchange-interaction network very complex by itself. Temperature-dependent changes in the magnetic structure, together with an abrupt emergence of a ferromagnetic component, are ascribed to continual spin reorientations described by a multicomponent, but yet unknown, spin Hamiltonian.