A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, $$\mathcal {E}$$ E , and a split disjunction, $$(l - x_j)(x_j - u) \le 0$$ ( l - x j ) ( x j - u ) ≤ 0 with $$l < u$$ l < u , equals the intersection of $$\mathcal {E}$$ E with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form $$\mathcal {K}\cap \mathcal {Q}$$ K ∩ Q and $$\mathcal {K}\cap \mathcal {Q}\cap H$$ K ∩ Q ∩ H , where $$\mathcal {K}$$ K is a SOCr cone, $$\mathcal {Q}$$ Q is a nonconvex cone defined by a single homogeneous quadratic, and H is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations $$\mathcal {K}\cap \mathcal {S}$$ K ∩ S and $$\mathcal {K}\cap \mathcal {S}\cap H$$ K ∩ S ∩ H , where $$\mathcal {S}$$ S is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.