For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V. We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.