The study focuses on structure analyses, texture analyses in particular, of an Al/Cu clad composite manufactured by single and double pass of the twist channel angular pressing (TCAP) method. Microscopic analyses were supplemented with numerical predictions focused on the effective imposed strain and material plastic flow, and microhardness measurements. Both the TCAP passes imparted characteristic texture orientations to the reinforcing Cu wires, however, the individual preferential grains’ orientations throughout the composite differed and depended on the location of the particular wire within the Al sheath during extrusion, i.e., on the dominant acting strain path. The second TCAP pass resulted in texture homogenization; all the Cu wires finally exhibited dominant A fiber shear texture. This finding was in accordance with the homogenization of the imposed strain predicted after the second TCAP pass. The results also revealed that both the component metals exhibited significant deformation strengthening (which also caused bending of the ends of the Cu wires within the Al sheath after extrusion). The average microhardness of the Cu wires after the second pass reached up to 128 HV, while for the Al sheath the value was 86 HV.