Serbinski Accounting Firms

OK, just to summarize:
If I have an employer specific work permit in Canada and I reside in Canada, then I can work for that employer as well as a contractor for US firm(s) without the US firms having to deal with me differently than they have when I resided in the US.
Is this correct?
Thank you.

Thank you for the information.
Is any of this impacted if the Canadian work permit is specific to a employer?

So, if each company withheld in its country, would that work (well)?

Separately, are there any tax implications of exchanging between currencies?

And to be an employee for a US company and a Canadian company (in different lines of business and both OK with it), would require that the US company have Canadian payroll services or could it be US based payroll for that employment (perhaps using a US address that I have)?
If contracting still ...

I am a US citizen.
I will be taking employment in Canada using a work permit.
I will also continue to do work in the US.
Based on what I have read, I should structure the work in the US as a contractor, correct?
Is there any other information or precautions that I need?
Thank you