There are two kind of offsets, a physical one (subtract DAC output before amplifier to ADC), and "display position" (simply shifts the waveform after acq).
For some scopes, it's the same - on the old TDS2xx for example, if you'd shift the waveform, it would change the physical offset. This way you'd always get the full dynamic range, even with a large offset. (I commonly measured ~1V of modulation on top of a 14V signal back then, so I would just move the waveform down).
For other scopes (most notably newer Tektronix ones), both are exposed individually. On the DPO7xxx-series you can press the position knob to switch between "position" and "offset". On some of the other scopes, you can only change position, but not offset using the knob (and you have to change the offset using the vertical menu).