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If you save it as JPEG that will normally produce an extra loss compared
to the original image.  However, if you have a JPEG which is exactly a
multiple of 16 in its X and Y dimension it can be losslessly rotated by 90
degrees.

Cheers,
David

Yes you have.  If you don't save it as a JPG, you've lost the space savings.
If you do re-save it as a JPG, you're going to re-run the compression algorithm.
That's not being run on the same (original, uncompressed) image data that was
the input to the first JPG compression, so it's extremely unlikely to come up
with the same final result.  That means you've lost some more image quality.
The same holds true even if you don't rotate the image; you can still end up
with a loss of quality just by re-saving a JPG image with no editing changes.
That's why some image editing software checks for this case, and will simply
copy the input file to the output in that situation.

Yes, you have, if you saved it as a JPEG.  You have re-compressed the image.
If you load a JPEG image, do something to it, and then save it again as a
JPEG, that is a lossy operation due to the compression.

However, a JPEG image with dimensions divisible by 16 can be rotated without
decompressing and recompressing the image at all -- that is, without ever
loading it into an image editor in the first place, requiring no further
step of re-saving as a JPEG and thus recompressing the data.

The normal case is that someone takes a bunch of pictures, loads them onto his
computer, looks at a screen full of thumbnails, and clicks a "rotate" button
to make the vertical compositions vertical.  If the saved images are JPEGs,
this can be done with no recompression loss if the image dimensions are
divisible by 16.

If you don't use JPEG, this doesn't matter to you.

You're right, it's because of compressing, but no you don't understand
it; it has nothing to do with semantics.

I'll have a stab at explaining it:

The original image comes out of the camera in JPG.  That is lossy
compression with effects; it has artifacts.  For illustration, let's
say that horizontal high-contrast areas cause a shadow in the lighter
region.  

You open that image (uncompressing it) in your viewer, and edit the
pixels.  What you're looking at has loss and artifacts.  You do a
pixel-for-pixel rotation (also duly rotating the artifacts). The
result in your editing window, as you point out, is exactly the
same as the original rotated.

Now, you save that edited image as JPG.  The saving process compresses,
and introduces new artifacts due to the new horizontal high-contrast
areas (as well as the previous, rotated artifacts of course).  This
compression loses more info.  When you open the newly formed JPG,
it no longer looks exactly like the original, rotated.  It's a lossless
rotation only so long as you don't save it in JPG.

On the other hand, lossless rotation is a mathematical operation on
the original JPG, effectively transforming it so that it uncompresses
into a rotated version.  There is no additional loss, no additional
artifacts.

Hope that helps?