A real \(n\times n\)-matrix A is strictly sign-regular iff all its minors are non-zero, and all \(k\times k\) minors have the same sign \(\sigma_ k(A)\), \(k=1(1)n\). Let \(A^*=DAD\), where \(D=diag(1,-1,...,(-1)^{n+1})\), and define \(A\leq^*B\) iff \(A^*\leq B^*\) entrywise. The matrix interval (relative to \(\leq^*)\) with smallest element \b{A} and largest element \(\bar A\) is \([A]=[\underline A,\bar A]\). The author proves that [A] is strictly sign-regular iff \b{A} and \(\bar A\) are strictly sign regular and \(\sigma_ k(\underline A)=\sigma_ k(\bar A)\), \(k=1(1)n\). He proves several related results, and conjectures that [A] is nonsingular and totally nonnegative iff \b{A} and \(\bar A\) are nonsingular and totally nonnegative.