Each piano string has two independent polarization modes doubling the coupled-mode count

Evidence

We have ignored another factor, namely string polarization. Each string can vibrate in two independent ways at the same frequency: (1) vertically, perpendicular to the plane of the soundboard, and (2) horizontally, parallel to that plane. An ideal, perfectly symmetrical string tied rigidly at both ends would have these two modes at exactly the same frequency. However, imperfections in the string and the fact that the bridge moves a little differently for vertical and horizontal vibration means that for three strings, there are not three but six possible modes available to drift in and out of phase with each other. — Eric J. Heller, Why You Hear What You Hear- An…, 1836