Bijective algorithm(s) that can shuffle a range of numbers back and forth
The subject is called format-preserving encryption,

and your last bullet point is addressed by tweaks.

Simple
http://web.cs.ucdavis.edu/~rogaway/papers/feistel.pdf

http://arxiv.org/pdf/1208.1176.pdf

http://web.cs.ucdavis.edu/~rogaway/papers/sr.html
Complex