The Fresnel–Arago laws by Augustin-Jean Fresnel and François Arago summarise some of the more important properties of interference between light of different states of polarization.
The laws are as follows (as per the translated version of the original report):
Two rays of light polarized at right angles do not produce any effect upon each other under the same circumstances in which two rays of ordinary light produce destructive interference.
Rays of light polarized in the same plane interfere like rays of ordinary light; so that in these two kinds of light the phenomena of interference are absolutely identical.
Two rays which were originally polarized at right angles may be brought to the same plane of polarization -without thereby acquiring the ability to interfere.
Two rays of light polarized at right angles and afterwards brought into the same plane of polarization interfere like ordinary light provided they were originally polarized in the same plane.
In the phenomena of interference produced by rays which have experienced double refraction the position of the interference bands is determined not only by difference of path and difference of speed, but in some cases, as above indicated, it is necessary to take into account also a difference of one-half a wavelength.
(5th law which was in the original report of experiments by Fresnel–Arago was left behind in later textbooks which may be due to the fact "Some cases, as above indicated" phrase is not well defined so readers may have to read the rest of their report in order to understand what these certain circumstances and cases were.)
When regarding the two constituent orthogonal linearly polarized states of natural light, we can rotate those (for example one component of natural light by sending it through a diagonal polarizer and an orthogonal polarizer in that exact order) to align with each other. So these rays would become parallelly polarized natural light waves with random to a degree phase fluctuations. But as the second Fresnel–Arago law states parallel natural light with phase fluctuations can form a stable visible interference pattern.
So why do two rays which were originally polarized at right angles when rotated to the same plane of polarization, not acquire the ability to interfere visibly (even slightly) despite their polarizations now being parallel? (Is there an explanation as to the change of the wave due to rotation or differences between original orthogonal polarized constituents of natural light that prevent visible interference completely when rotated into parallel polarization?)
Note: This question has been edited several times in light of later revealed facts and for further clarification of the question. However, it is still firmly grounded within the scope of the original query. &Thanks to Farcher for providing Fresnel–Arago original paper.