BiFeO$_3$ thin films were epitaxially grown on (110)- and (001)-oriented NdGaO$_3$ single crystal orthorhombic substrates by pulsed laser deposition. The films grown on NdGaO$_3$(110) are fully strained and show two ferroelectric variants that arrange in a stripe domain pattern with 71° domain walls, as revealed by piezoresponse force microscopy. We explored their antiferromagnetic textures using scanning NV magnetometry. Surprisingly given the large compressive strain state, the films still show a spin cycloid, resulting in a periodic zig-zag magnetic pattern due to the two ferroelastic variants. The films grown on NdGaO$_3$(001) are also fully strained, but the (001) orthorhombic substrate imposes a strongly anisotropic in-plane strain. As a consequence, the ferroelectric polarization exhibits a uniaxial in-plane component, parallel to the b-axis of the substrate. The ferroelectric domain pattern consists of 109° charged domain walls between the two selected ferroelastic variants. This anisotropic strain impacts the magnetic state of BiFeO$_3$ and leads to a simpler spin texture defined by a single propagation vector for the spin cycloid. In both cases, electric-field control of ferroelectric domains tends to favor a transition to a canted antiferromagnetic order. These results reveal that the cycloidal structure of BiFeO$_3$ can undergo large compressive strain and open further electrical means to tune the magnetic state of this room-temperature multiferroic compound.