What makes you think that the system which displays the GERMAN cross will show a different indication to the control deflection? If there is an error, it will affect both presentations in the same sense.
OK. Let us suppose the output of the sidestick is sent on a data bus or wires both to the DMC's, which generate the white crosses, AND the flight control computers, which of course move the controls. However, let us suppose that either the DMC or a flight control computer has a fault
internal to itself, (for example a dry joint)*, which reverses the sense of the sidestick - hey presto, one system thinks 'stick left' the other thinks 'stick right'. The box appears after engine start (i.e. before taxying), and not on application of take-off power - why would that be if it was not to provide sidestick position info during taxying?
That's my logic for my extra check, (which by the way takes all of half a second to perform). From what I know about electronic circuits and dry joints etc., this is what I will continue to do.
The OP wanted to know what the white cross was used for. I think that myself and others have answered their question.
U
*For those who may not know; a dry joint is a faulty electrical soldered connection in a circuit, which results in an intermittent electrical connection. Sometimes the joint is electrically sound, other times it is open circuit. These joints can and often do look perfect to the naked eye, and when electrically tested can appear sound. However, when the circuit heats up to operating temperature the joint can go intermittent. Dry joint faults are very difficult to trace.