You can convince yourself of this with your data:

	A(X)	B(Y)
1	-5	0
2	-4	0
3	-3	0
4	-2	1
5	-1	1
6	0	1
7	1	1
8	2	1
9	3	0
10	4	0
11	5	0

Plot your data as a Line+Symbol plot and select Analysis:FFT. Click the Settings Tab of the FFT tool and UNCHECK the Normalize Amplitude and Shift Results options. (This is appropriate for our planned Inverse FFT.)

Click the Operation Tab and select Forward and Amplitude before clicking OK.

Minimize or Close the FFTPlot1 to reveal the FFT1 worksheet. As you noted, the Imag(inary) data is non-zero. Let's now do an Inverse FFT - first using just the Real data, then using both Real and Imag(inary) data.

Inverse FFT with Real data

Highlight column Real of the FFT1 worksheet and select Analysis:FFT. Uncheck the two options on the Settings Tab as before. Note that the Imaginary field on this Tab is blank. Click the Operation Tab and this time select Backward and Amplitude before clicking OK.
Note that this gives two 'pulses' where your original data only had one.

Inverse FFT with Real and Imaginary data

Highlight both columns Real and Imag of the FFT1 worksheet and select Analysis:FFT. Uncheck the two options on the Settings Tab as before. Note that the Imaginary field on this Tab now uses that Imaginary data you thought should be zero. Click the Operation Tab and select Backward and Amplitude before clicking OK.
Note that this gives your original data (plus the few zero-valued points that were added to make the Fourier Transform a Fast Fourier Transform).

While giving you only real results might be correct for real data, I think you can see the advantages of reporting the imaginary data as well.