swapfuck
swapfuck is a brainfuck derivative by User:Rdococ, which replaces the typical increment and decrement instructions with a single 'swap' instruction.
Differences from brainfuck
 Input is loaded into the tape at the beginning of execution.
 Cells can only hold a
0
or a1
.  The only instruction that modifies the tape is the 'swap' instruction, which swaps the contents of the current cell with the cell to the right.
Instruction set
Instruction  Description 

@ 
Swaps the values of the current cell and the cell to the right. 
< 
Move the cell pointer to the left. 
> 
Move the cell pointer to the right. 
[ 
Jump past the matching ] if the current cell has a value of 0.

] 
Jump back to the matching [ if the current cell has a nonzero value.

. 
Adds the bit at the current cell to the output queue. When the output queue reaches a size of 8, it is emptied and the corresponding UTF8/ASCII character is printed. 
Computational class
If the input tape contains sufficient consecutive pairs of 1 0
, a swapfuck program could operate on each pair as if it was a single, flippable bit.
If swapfuck's []
loops were replaced with ones that looped while the current cell was zero, not nonzero, a finite input tape of 1 1 1 1 1 1
would allow a program to simulate threecell unbounded brainfuck, which is Turing complete. The tape is split into 6, interleaved bit streams  the first three represent the values of the cells, and the last three are used simply to find the beginning of the tape while remembering the current cell position.
Performing Turing complete computations in regular swapfuck is more difficult, but likely not impossible. The tape could be split into 3 + n interleaved bit streams, where n is the number of bits required to keep track of the current register in use and invert its value for seeking. Then, we only need a finite number of 1s for this bookkeeping, as we can lug them around with intensive swapping.
Implementation
A rough implementation of swapfuck in Lua is available here.