Do comercial spectrum analyzers show a single-sided spectrum?
There are many types of SA's but all that are designed to operate in your carrier range will detect your SSB as expected.

Ignoring FFT's for now, the most common SA's have an extremely wide sweep range. The input channel is up converted from fmin ~ fmax with a fixed Local oscilator such that f >> 2fmax . The internal sweep VCO now only has a smaller tuning ratio.

The Lower sideband is passed onto a 2nd IF channel where you can tuned the BW and resolution of the AM detector LPF.

Now your spectrum is rendered accurately for scalar values with the spectral density function using a 6~10 decade log amp where noise depends on BW and LPF resolution of the detector as well input attentuation and LO drive levels control.

It operates much the way most UHF radios work that have a wide tuning range for AM.  YOu can detect FM and PM by using the skirt of the 2nd IF BPF slope. For vector spectrum special detectors are employed to display PM and AM in the typical constellation after center f is tuned.  YOu can even watch SSB TV signals with any wire near a window. 

So a Spectrum Analyzer (SA) is a smart radio tool to analyze your signals and capable of rendering them accurately. 

There are also FFT SA's, VNA's and many other types.

Thank you for letting me assist you in this explanation, so you can understand how it works like a radio.