A rather tenuous connection, but I'll throw it out and hope somebody can build a better answer from this (in short, too long for a comment):
It is well known (or it should be!) that one can use the Sturm sequence construction for a polynomial $p(x)$ with all roots real and its derivative to construct a symmetric tridiagonal companion matrix (that is, the symmetric tridiagonal matrix whose characteristic polynomial is $p(x)$). See Fiedler's paper for details.
One could then apply Cauchy's interlacing theorem on the tridiagonal matrix just constructed. As a matter of fact, the characteristic polynomials of successive leading submatrices of a tridiagonal form a Sturm sequence.
To expand on the comment I gave regarding orthogonal polynomials: consider the monic orthogonal polynomials $p_n(x)$ satisfying the difference equation
One can associate a symmetric tridiagonal matrix with this recursion, called a Jacobi matrix:
whose characteristic polynomial is $p_n(x)$. It can be seen that the characteristic polynomials of the leading $1\times1, 2\times 2, \dots$ submatrices are $p_1(x),p_2(x),\dots$ We see here the correspondence between the Sturm sequences for orthogonal polynomials and tridiagonal matrices.