How to rotate a jpg without losing format?

If you rotate an image, then there are three problems which may affect the image size.
1) If you do not rotate it to a width and height which are the same as the original. For example, if you rotate a square image by 45 degrees, the resulting image needs a bigger square in which to sit - and all images are based on a rectangular box.
2) Jpeg files use a lossy compression - when you rotate an image by any amount, you may well end up with an image which does not compress as well. This can make the file significantly bigger.
3) When you write a jpeg file, the compression amount can be adjusted. If the routine used to save the image is not the same as the original image was saved with, or if the compression parameters are different in any way, teh resulting file will not be the same size.

Generally speaking it is a bad idea to do any image processing on a jpeg file (including rotating it). Jpeg is a lossy compression algorithm, so any sbsequent processing is already working on a degraded image. Work directly from an original bitmap or other non-lossy compressed format) if you can, and save jpegs from that as separate files.