You are an employee of the travel company and paychecks come from them. Taxes are determined based on your legal residence, so lucky you! I believe the tax rules change if you stay at an assignment for 6 months or more.
I have found that most travel companies don't know the tax code. For example, they will allow you to claim the full IRS per diem amount (which includes meals, housing and incidentals) even though you are being reimbursed for housing...double dipping as far as the IRS is concerned. I would advise you to meet with a CPA/tax advisor and find out what is legal and what is not. It may cost you a little but is well worth it.