Installation for manufacturing a reinforcing ply (2, 2_left, 2_right), said installation comprising: - a preparation station which is arranged to prepare strips by cutting a band called "straight wire band" which is formed by a plurality of reinforcing wires which are embedded in at least one layer of rubber and which extend parallel to one another in the longitudinal direction of said straight wire band, - a jointing station arranged to successively butt-joint, by their selvedges (5L, 5R), the strips obtained from the preparation station so as to form a reinforcing ply (2, 2_left, 2_right), - a station for turning over strips, which is interposed between the preparation station and the jointing station and which comprises: - a first receiving surface, arranged to receive a strip prepared by the preparation station and to hold said strip according to a first orientation in which said strip, which is delimited in thickness by a first face (5_1) and by a second face (5_2) opposite said first face (5_1), rests and is held on said first receiving surface by its second face (5_2), and presents its first face (5_1) as visible face, - a second receiving surface associated with a reversing device which is arranged to transfer a strip, initially placed and held on the first receiving surface, from said first receiving surface to said second receiving surface, thus turning over the strip so as to confer on said strip a second orientation, the reverse of the first orientation, in which said strip is held against the second receiving surface by its first face (5_1), and presents its second face (5_2) as visible face, said installation being characterized in that it comprises - a gripping member arranged to take a strip selectively from the first receiving surface or from the second receiving surface in order to allow the jointing station to selectively construct a reinforcing ply (2, 2_left, 2_right) from strips oriented in accordance with the first orientation or, respectively, in accordance with the second orientation. - a preparation station which is arranged to prepare strips by cutting a band called "straight wire band" which is formed by a plurality of reinforcing wires which are embedded in at least one layer of rubber and which extend parallel to one another in the longitudinal direction of said straight wire band, - a jointing station arranged to successively butt-joint, by their selvedges (5L, 5R), the strips obtained from the preparation station so as to form a reinforcing ply (2, 2_left, 2_right), - a station for turning over strips, which is interposed between the preparation station and the jointing station and which comprises: - a first receiving surface, arranged to receive a strip prepared by the preparation station and to hold said strip according to a first orientation in which said strip, which is delimited in thickness by a first face (5_1) and by a second face (5_2) opposite said first face (5_1), rests and is held on said first receiving surface by its second face (5_2), and presents its first face (5_1) as visible face, - a second receiving surface associated with a reversing device which is arranged to transfer a strip, initially placed and held on the first receiving surface, from said first receiving surface to said second receiving surface, thus turning over the strip so as to confer on said strip a second orientation, the reverse of the first orientation, in which said strip is held against the second receiving surface by its first face (5_1), and presents its second face (5_2) as visible face, - a gripping member arranged to take a strip selectively from the first receiving surface or from the second receiving surface in order to allow the jointing station to selectively construct a reinforcing ply (2, 2_left, 2_right) from strips oriented in accordance with the first orientation or, respectively, in accordance with the second orientation.