Reversible hash function?
What you are asking for is encryption.  A block cipher in its basic mode of operation, ECB, reversibly maps a input block onto an output block of the same size.  The input and output blocks can be interpreted as numbers.
For example, AES is a 128 bit block cipher, so it maps an input 128 bit number onto an output 128 bit number.  If 128 bits is good enough for your purposes, then you can simply pad your input number out to 128 bits, transform that single block with AES, then format the output as a 128 bit number.
If 128 bits is too large, you could use a 64 bit block cipher, like 3DES, IDEA or Blowfish.
ECB mode is considered weak, but its weakness is the constraint that you have postulated as a requirement (namely, that the mapping be "deterministic").  This is a weakness, because once an attacker has observed that 123 maps to 9874362483910978, from then on whenever she sees the latter number, she knows the plaintext was 123.  An attacker can perform frequency analysis and/or build up a dictionary of known plaintext/ciphertext pairs.
I need a reversible hash function (obviously the input will be much smaller in size than the output) that maps the input to the output in a random-looking way. Basically, I want a way to transform a number like "123" to a larger number like "9874362483910978", but not in a way that will preserve comparisons, so it must not be always true that, if x1 > x2, f(x1) > f(x2) (but neither must it be always false).
The use case for this is that I need to find a way to transform small numbers into larger, random-looking ones. They don't actually need to be random (in fact, they need to be deterministic, so the same input always maps to the same output), but they do need to look random (at least when base64encoded into strings, so shifting by Z bits won't work as similar numbers will have similar MSBs).
Also, easy (fast) calculation and reversal is a plus, but not required.
I don't know if I'm being clear, or if such an algorithm exists, but I'd appreciate any and all help!