When an Immigration Officer (IO) determines that an leave-to-enter is to be refused or cancelled, he explains the situation to somebody in his reporting line and gets their agreement. Normally this will be a Chief Immigration Officer. An "Immigration Officer" is any person designated by the Home Secretary to act in that capacity and who has taken an oath to serve HM. On the rare occasion you can see a Police Constable temporarily acting in that capacity.
The rules also provide for an Immigration Inspector to get involved, but I have never seen it happen.
These appear to be conflicting rules.
Define 'conflicting'. The rules (as opposed to laws) are a product of distilling about 35+ various acts of Parliament into a contiguous stream of text that people can refer to without having to wade through a mountain of legislation. The process invites ambiguity and people on both sides (i.e., the person and the IO's) benefit from it. We like it. If a piece of ambiguity becomes intolerable, the Tribunal can step in and provide a clarification. And if a Tribunal's decision does not work in a given situation, the Upper Tribunal can step in and help out.
Can an immigration office cancel leave to remain, or not?
The IO will need permission from a CIO before cancelling leave to remain. If it is at a port of entry Paragraph 10A is the controlling reference point.