The Linistepper generates 6 microsteps in hardware, using the diodes and resistors to make the correct voltage levels for the 6 microsteps. This gives 200 x6 = 1200 steps/rotation which is one of the Linistepper microstep modes that you can select.
To make 3600 steps/rotation the Linistepper alternates very fast in software between 2 of the microsteps. So it either does AAA or AAB or ABB (where A and B are 2 sequential microsteps) and at a high frequency around 333 kHz if you are using the 16MHz xtal. The overall effect is to provide 3600 microsteps from the 1200 hardware microsteps.
Interesting note; the Linistepper is the only stepper driver that produces 3600 steps/rotation, making it ideal for things that work in degrees as it makes 10 microsteps per degree, also the smoothing caps C5 and C6 can be tuned (by choosing their value) to give very smooth transition from one microstep to the next when running at the "tuned" motor speed. Both those factors make it ideal for telescope movement (astronomy etc) and pan/tilt setups etc.