>How big is the subset of QR codes that are valid in more than one orientation?
Given the placement of the position and alignment markers in a valid QR code it looks like the only symmetry it can have is reflection (mirror-image) symmetry about its main diagonal (the line from the top-left to the bottom-right) [1]. If you take any given QR code and replace the part of it that is below the main diagonal with a mirror image of the part above it then in the worst case you will lose 50% of the information contained in the code (actually less since the pixels on the diagonal stay there no matter what). Now, the content of a QR code generated with the heaviest error correction settings can be recovered even if up 30% of its data is lost [2]. If you generate a QR code where the part above the main diagonal is similar enough to the one below it (meaning at least 50% - 30% = 20% of symmetrically located pixels are the same color) then such a QR code should survive being made fully symmetrical.
The above is a simplification, though, since, generally speaking, error correction in QR codes operates on blocks and not whole codes. In practice it also may or may not be possible to generate a sufficiently symmetrical code in the first place, let alone one with the specific content you want.
[1] See http://imgur.com/YMNIsgc. The red line show the diagonal, the green arrows point out pairs of symmetrical pixels.
Given the placement of the position and alignment markers in a valid QR code it looks like the only symmetry it can have is reflection (mirror-image) symmetry about its main diagonal (the line from the top-left to the bottom-right) [1]. If you take any given QR code and replace the part of it that is below the main diagonal with a mirror image of the part above it then in the worst case you will lose 50% of the information contained in the code (actually less since the pixels on the diagonal stay there no matter what). Now, the content of a QR code generated with the heaviest error correction settings can be recovered even if up 30% of its data is lost [2]. If you generate a QR code where the part above the main diagonal is similar enough to the one below it (meaning at least 50% - 30% = 20% of symmetrically located pixels are the same color) then such a QR code should survive being made fully symmetrical.
The above is a simplification, though, since, generally speaking, error correction in QR codes operates on blocks and not whole codes. In practice it also may or may not be possible to generate a sufficiently symmetrical code in the first place, let alone one with the specific content you want.
[1] See http://imgur.com/YMNIsgc. The red line show the diagonal, the green arrows point out pairs of symmetrical pixels.
[2] http://archive.is/20120915/http://www.tec-it.com/de/support/...