From the pair (nth Fibonacci number,(n-1)th Fibonacci number) you can get the next pair by multiplying by the matrix

1 1

1 0

This matrix has inverse

0 1

1 -1

which you can use to find the "previous pair" (so for the one of interest, you take the difference of the current pair, as you say).

The first matrix is helpful in analysing the behaviour of the recursion rule on any starting pairs. There are two Perron-Frobenius eigenvalues. There's a leading or "expanding" direction, with eigenvalue value the golden mean, and a contracting direction with eigenvalue value -1/golden mean. That means almost all pairs converge towards the expanding eigenline, where the ratio converges to the golden mean. However, you can also take pairs on the contracting eigenline, where the ratio of successive terms is always exactly -1/golden ratio and they converge to 0.