We extend our previous work on structural steganalysis of LSB replacement in digital images, building detectors which analyse the effect of LSB operations on pixel groups as large as four. Some of the method previously applied to triplets of pixels carries over straightforwardly. However we discover new complexities in the specification of a cover image model, a key component of the detector. There are many reasonable symmetry assumptions which we can make about parity and structure in natural images, only some of which provide detection of steganography, and the challenge is to identify the symmetries a) completely, and b) concisely. We give a list of possible symmetries and then reduce them to a complete, non-redundant, and approximately independent set. Some experimental results suggest that all useful symmetries are thus described. A weighting is proposed and its approximate variance stabilisation verified empirically. Finally, we apply symmetries to create a novel quadruples detector for LSB replacement steganography. Experimental results show some improvement, in most cases, over other detectors. However the gain in performance is moderate compared with the increased complexity in the detection algorithm, and we suggest that, without new insight, further extension of structural steganalysis may provide diminishing returns.