"For simple non-live-shape paths it won’t be applicable anyway, same as rotation, but live shapes would definitely benefit from storing it."

Currently it can rotate an ordinary shape and record the rotation just fine so it is applicable. But Shear does not work for ordinary shapes or live shapes. My post wasn't for live shapes, my post was for every kind of object. Shear doesn't work for live shapes, ordinary shapes, or raster images but rotate works for all of them.

Larry, the Shear is not supported as an inherited value, nor by path or even live shapes (rectangles, ellipses, stars, etc.), so the values is not stored within it and you can’t input one, see it in the field, and then change it back to zero, like we can do with angle value for live shapes. Shearing is a 'final deformation'. Therefore each mouse wheel rotation applies 1° and it keeps shearing the changes path again and again, rather than changing the original shape.
So I’d call it not a bug, but rather a request to support shearing for live shapes. For simple non-live-shape paths it won’t be applicable anyway, same as rotation, but live shapes would definitely benefit from storing it.

If you make an object, then place the cursor in the Shear box in the Transform panel, then use the up/down arrow, or roll scroll wheel, the object will shear as expected, but the readout will flicker from 0 to 1 degree, then back to zero. If you keep changing the number, it continues to flicker. If I hold Shift and use the arrow key, it will sometimes flash 10 degrees, then back to 0 degrees. If I type a number in the box, the object will respond but the box immediately shows 0 degrees again.

I confirm.
Rectangle (being a path with a custom 'live shape' attribute, allowing it to have rectangle-specific control in Transform and elsewhere) looses the 'live shape' status when a Shear value is applied to it.
It looks like these just don’t support shearing.

NOTE: if you put the opposite shear value of what you set your shear to, it WEIRDLY does NOT actually set it back to original. It's off ever so slightly. E.g. if you shear a rectangle -15 degrees then shear it 15 degrees, your lines won't be straight. It's maddening.

InDesign shearing is superior because you can see where an item's angle is and change it at any time. The best part, however, is that you can extend the sides or resize the sheared shape and it retains the correct angle, allowing you to reuse assets much more easily.

While rotating objects keeps the value of the angle in the dialogue box, bending (shearing) immediately forgets the value. A bug?

Während beim Drehen von Objekten der Wert des Winkels im Dialogfeld erhalten bleibt, vergisst das Verbiegen den Wert sofort wieder. Ein Bug?

See screenshort in German:

With the Shear Tool selected, you can Option-Click on the target, which pulls up a dialog that allows you to type in the value. You'd just make the value the opposite of positive or the opposite of negative. What we need is a way to shear objects in a non-destructive manner.

Using the shear in the transform window works but not correctly, When using an arrow to shear an item, the number switches from 0 degrees to 1 degrees and then back to 0 degrees and does not hold the changed degrees number but does shear the item. so if you want to shear say 15 degrees you can push the arrow key 15 times and it will work or you can type in 15., BUT will revert back to 0 degrees so you have no clue how much it is sheared. Rotation degrees works fine in the same transform window. Attached is a video showing the problem

When I transform an object, the rotation angle remains visible in the Transform panel, while that the inclination angle disappears... why? it's very frustrating...
In InDesign this is possible and correct.... not in Illustrator.

It is not possible to reset or reproduce the angle of inclination on another object.

In the Transform Panel I use the mouse wheel to adjust the Shear of an object but it does not show the degrees that it is moving. The object does adjust but the panel only reports 0°. As I scroll with the mouse wheel I can see the number flickering. When I click on the selection arrow I can select a number but that only adds to the number of degrees instead of telling me how many degrees I have adjusted it. I have attached a short video where I am adjusting the shear using the mouse wheel

good idea!

This is a great suggestion. I use Autodesk Maya and storing an objects history would be useful! I find in Illustrator i keep multiple versions of a created object so if something doesnt work out i can delete it and use a previous version of the shape. Keeping the history in some way practical sense is worth exploring!

It's kind of weird idea. I understand why you want this. But 'Object' is not a unchangeable thing. I can cut segments, join them, move anchors — how would this reflect on 'object's' shear/rotation/scaling? What if I take rectangle (path in fact) and manually move it's anchors away from center, making it larger without scaling it? (I do this more often then scaling) And what if I try to 'reset scaling' then?
When I wish to control rotation after some time, I use 'Transform' effect. A lot of then, stacked. I also wish that I could combine interactivity of Rotate toll with an ability to edit rotation automatically. Like Some option for a toll would create this Effect for me. This can be done.
But your idea is just not that well-formed yet. You have my vote still.

Or, alternatively:

It would be cool if any kind of transformation would always be available as a separate entry in the Appearance panel, and thereby removable/editable.

The programming effort for Adobe to show these transformation properties in the Appearance panel and allow them to be reset to "0" or to be edited to any other value by entering decimal values would be minimal, as this internally already happens, just without having the Appearance GUI provided yet.

Would be great if you could command click on the transform icons to reset, as in the character palette. Shear would be great as a live effects added to the transform effect. The free distort effect is quite jumpy and not very precise, as sometimes I need to repeat the distortion on different sized elements. This is a great suggestion, and especially shearing needs updating.