The object of this paper is to study certain joint cobordism groups \(C^ 0(n,k;\ell)\) of immersions defined by F. Uchida [ibid. 8, 207-218 (1971; Zbl 0222.57017)]. Elements are bordism classes of completely regular immersions \(f: M^ n\to N^{n+k}\) without \((\ell +1)\)-tuple points; both M and N are oriented and may vary up to oriented bordism. A Pontryagin-Thom type argument identifies \(C^ 0(n,k;\ell)\) with \(\Omega_{n+k}(\Gamma_{\ell} MSO(k))\), where \(\Gamma_{\ell}(-)\) denotes the \(\ell\)-th step in the Barratt-Eccles functor [M. G. Barratt, P. J. Eccles, Topology 13, 25-45, 113-126, 199-207 (1974; Zbl 0292.55010, Zbl 0292.55011, Zbl 0304.55010)] As a consequence, the natural maps \(C^ 0(n,k;\ell -1)\to C^ 0(n,k;\ell)\) are shown to be monomorphic and the groups \(C^ 0(n,k;\ell)\) are calculated rationally.