Can a JPEG compressed image be rotated without a loss in quality?

Yes, it is possible for certain cases: 90-degree rotations and flips on images with dimensions that are a multiple of 8. The heart of the JPEG algorithm -- the lossy part -- involves breaking the image into 8x8 pixel blocks, performing a discrete cosine transform on the block and then quantizing the result. There's also some color space conversion and lossless compression of the blocks on top of this.

Absolutely - just change the orientation value in the EXIF data. The vast majority of image programs will respect this setting and show the picture "rotated".

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