Yes. This is due to 2nd order diffraction. Similarly, you would observe 3rd signal at 960 nm. The intensity will be progressively weaker and needless to say it is 320 nm signal (not 640 or 960 nm). Use a filter to block higher order signal. Spectrometers such as Varian have inbuilt filters (called "auto filter") which can be used to block such signal.
I am not entirely sure what you are trying to mean by "both excitation and emission wavelengths". If you see a signal at x (irrespective of whether it is scattering or emission) you will also see signal at 2x, 3x,.... If you excite the sample at 300 nm it will have Rayleigh scattering signal at 300 nm. So following the same logic it will have signal at 600 nm also.
What Tuhin has said is correct that when you observe emission (in the range 320-700) under 300 nm excitation. You will observe the double band signal at 600 nm and third order band signal at 900 nm. Similarly while recording excitation spectra (in the range 200-600 nm) for suppose 620 nm emission, you will get double band at 310 nm. To cut this signal from your emission or excitation spectra filters with suitable wavelength can be used.
Kindly measure the emission spectra again from 300-700 nm. This time use regular transparency sheet ( which has cut off wavelength 310 nm) as a filter. With this you will not get 600 nm double band.