The paper deals with the regular indefinite Sturm-Liouville eigenvalue problem \[ -f''+qf=\lambda rf \text{ on }[a,b],\tag{1} \] subject to the self-adjoint boundary conditions \[ C\left(\begin{matrix} f'(a)\\-f'(b)\end{matrix} \right)=D\left(\begin{matrix} f(a)\\f(b)\end{matrix} \right),\tag{2} \] where \(C,D\in \mathbb{C}^{2\times 2}\) satisfy the conditions \(\text{rank}(C|D)=2\), \(CD^*=DC^*\), \(q\) and \(r\) are real integrable functions on \([a,b]\), \(r\not=0\) a.e.on \([a,b]\), and \(r\) changes sign at finitely many points in \((a,b)\) (called turning points). The authors give sufficient conditions, and necessary and sufficient conditions on the coefficients \(q,r,C\) and \(D\) under which there exists a Riesz basis for the Hilbert space \(L^2_{|r|}[a,b]\) consisting of root functions (i.e., eigenfunctions and associated functions) of problem (1)–(2). A generalization of Parfenov’s criterion to certain classes of odd-dominated weights is also presented. The authors prove that the Riesz basis property of (1)–(2) depends only on the local behavior of \(r\) in neighborhoods of the turning points if and only if the boundary conditions \((2)\) are not row equivalent to the boundary conditions \[ e^{it}f(a)=f(b),\;f'(a)=e^{-it}f'(b)+df(a).\tag{3} \] An application of equation (1) with the boundary conditions (3) to a regular HELP-type inequality without boundary conditions is finally addressed.