Is there a standard term for monitors which can be adjusted to both landscape and portrait? Do they signal the computer when it changes?

I'm thinking of upgrading my monitor. I've noticed that flat panels that can be adjusted between landscape and portrait mode are no longer always expensive and not necessarily scarce.

But searching for them to do price and feature comparisons seems hard because most sites for my local electronics shops don't list this feature in their search page and even descriptions of the monitors don't seem to always mention they can do this, or state it in a variety of ways, most of which are ambiguous.

Unless I'm missing something.

You could describe this feature as "tilting", "rotating", "pivoting", or "swivelling", but all of those terms could just mean that there's some degrees of adjustment of the viewing angle around the X, Y, and/or Z axes. Or just "portrait mode". Is one of these actually more or less standard or at least less ambiguous?

Related, do such monitors signal the OS when you twist it between landscape and portrait? If only some of them do so, does this feature have a name? (If it depends on the connector type etc I suppose it's getting off-topic so just the standard way to describe this signalling feature will do.)

You haven't missed anything - the terminology changes between hardware companies, and many review sites do not do a complete job. However, most manufacturers do a pretty decent job of describing their products.

You must distinguish between the purely mechanical part and the software part (the driver).

Changing the monitor's orientation must be done by the video driver, or the results would be unusable with text that is badly oriented and unreadable without some neck gymnastics. I haven't seen monitors that have such a setting, but such monitors must have some way of signalling their proprietary drivers of the change. Most drivers I have seen have this option in their settings, using terms such as (among others) Orientation, Landscape, Portrait, Flipped.

Mechanical manipulations are independent of the video driver. The most prevalent are the terms you have already listed: tilting, rotating, pivoting, or swiveling, but there might be others.

I just Googled "portrait monitors" and "vertical monitors" and got lots of hits.

As for how the orientation is set, I am aware of only a couple of monitors that have this functionality. In my experience, it is a case of setting the orientation in your computers display settings.

EDIT: To add, pretty much any monitor can be used in portrait mode if you mount it on a VESA mount/arm. There are arms which allow you to rotate the monitor at will, or you can just mount the monitor on the VESA mount in portrait mode. I have seen people mounting ultrawide and curved monitors vertically using a VESA mount.