The zeroth-order component of the cosine expansion of the projected three-point correlation function is proposed for the clustering analysis of cosmic large-scale structure. These functions are third-order statistics but can be measured similarly to the projected two-point correlations. Numerical experiments with N-body simulations indicate that the advocated statistics are redshift distortion free within 10 per cent in the non-linear regime on scales of ∼0.2–10 h−1 Mpc. The halo model prediction of the zeroth-order component of the projected three-point correlation function agrees with simulations within ∼10 per cent. This lays the ground for using these functions to perform joint analyses with the projected two-point correlation functions, exploring galaxy clustering properties in the framework of the halo model and relevant extensions.