Why does the spacing become uneven in these shapes after rotation?

You are looking at a basic geometry and real-world example of the difference between aligning / distributing centroids versus aligning / distributing edges in a non-linear group – it’s not the software – this is just how reality in a Euclidean geometric reference frame works.

Were your lozenges centre-aligned horizontally as well as distributed between centroids, giving each an equal individual rotation would result in equal horizontal spacing; however, as they are aligned centre-top-centre, this cannot occur – see quick diagram.

The quick takeaway is that the object centroids are still equally spaced: due to the vertical alignment shift, this cannot still be true of the shape’s borders; if they were circles, all relative distances would be maintained, as in a circle the horizontal distance from centroid to edge is always the same no matter the rotation applied.

Edit post question being edited and changed in specific requirements:

Illustrator’s Transform Each will work exactly as needed for your chosen use case, retaining top alignment without issue:

And adjunct to seeing @Scott’s comment to my answer, I have added two more clarifying diagrams – Transform Each works as expected, as Scott had initially answered.

First, Transform Each being applied with Preview on:

And subsequent dimensioning of the resultant with appropriate annotation – my earlier answer, combined with Scott’s absolutely correct recommendation of Transform Each gets a reliable, logical result – yay!

Hope this helps.