Technically they do have the right to request proof of employment in previous positions.
In the Guidebook they state: "Photocopy of proof of academic qualifications and relevant work experience".
They haven't insisted on that for any of my applications for several years now but there were instances in the past. It may be related to lack of academic qualifications and therefore requiring more work experience proof, or other reasons, I never really analysed what was behind the request.
I think best advice is if you have them, get them ready. If you don't have anything to prove declared previous employment, don't fret yet. Most companies give a record of service, with no mention of reason for leaving or specific performance information, which is really all they want. As others have said, they do not need to know you were terminated, resigned or walked off the job, only the period of the employment and position(s) held.