In this paper, a backscatter cooperation (BC) scheme is proposed for non-orthogonal multiple access (NOMA) downlink transmission. The key idea is to enable one user to split and then backscatter part of its received signals to improve the reception at another user. To evaluate the performance of the proposed BC-NOMA scheme, three benchmark schemes are introduced. They are the non-cooperation (NC)-NOMA scheme, the conventional relaying (CR)-NOMA scheme, and the incremental relaying (IR)-NOMA scheme. For all these schemes, the analytical expressions of the minimum total power to avoid information outage are derived, based on which their respective outage performance, expected rates, and diversity-multiplexing trade-off (DMT) are investigated. Analytical results show that the proposed BC-NOMA scheme strictly outperforms the NC-NOMA scheme in terms of all the three metrics. Furthermore, theoretical analyses are validated via Monte-Carlo simulations. It is shown that unlike the CR-NOMA scheme and the IR-NOMA scheme, the proposed BC-NOMA scheme can enhance the transmission reliability without impairing the transmission rate, which makes backscattering an appealing solution to cooperative NOMA downlinks.