Pixels blur/ distort when rotating image

In the original example you posted, you haven't just rotated, you have also rearranged the elements relative to each other. So that suggests you have performed repeated transforms - in other words, cumulative resampling.

Every time tou resample the pixels, it gets slightly softened. That's just because it's made of individual pixels.

To minimize this, use smart objects. You will still get one final resampling when done, but you avoid the cumulative effects. A smart object always contains an unchanged reference original embedded in the file.

You can also try different resampling algorithms. Note that this can be set individually for many tools, which will override the preference setting.

Oh, and as Bojan suggests, always view at 100% to judge the result. This maps exactly one image pixel to exactly one screen pixel, so you get a true representation on screen of the full pixel data.