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The trace on my Tektronix scope is rotated (to align it with the internal graticule) by varying the DC in an axial solenoid that the beam passes through. I don't know how fat that technique can twist the scan without distorting it, but if it can carry you to +/-22.5 degrees, that might be enough.
My preferred way: The deflection currents are analog. Matrixing the nominal X and Y drives will swing the beam to any angle you want. What you need to do that is a pair of digital volume controls -- one fed with X and a digital gain, the other with Y and a different digital gain -- between the DACs and each deflection driver. Computing the gains is a one-time thing. If you don't like the math, you can use a LUT.
The software to rotate the scan with the drive coils uses the same algorithm used for drawing slanted lines on stepper-motor plotters. Once thought through, implementation is straightforward.
45 degrees is trivial, but there is a sqrt(2) scale factor that you might need to correct.Think Moire. In any case, if your step size exceeds your resolution limit, the image will be (non alcoholically) pixellated.
Clear as is.
See above. Great minds and all that.
I hope I helped.
Jerry