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

$$p_{n+1}(x)=(x-c_n)p_n(x)-d_np_{n-1}(x),\qquad p_{-1}(x)=0,p_0(x)=1$$

One can associate a symmetric tridiagonal matrix with this recursion, called a Jacobi matrix:

$$\begin{pmatrix}c_0&\sqrt{d_1}&&\\\sqrt{d_1}&\ddots&\ddots&\\&\ddots&&\sqrt{d_{n-1}}\\&&\sqrt{d_{n-1}}&c_{n-1}\end{pmatrix}$$

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.