What does your contract provide? Generally if a company, lets say Big Box, hires your company to construct a building, the contract sets the terms of payment. If it is a fixed price contract, they generally would not look at the subcontracts. However, if it is cost plus, then they may have the right to audit and to confirm. There may be other circumstances where they have the right to review the invoices. You need to discuss your situation and your business model with a construction attorney. You may also need to consider what happens if they decide to manage their own construction project (assuming there are not licensing issues).
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If you have a written contract with that company, its terms will govern as far as the form of supporting documents you need to get paid. They can put any of these sorts of requirements into the contract and if you sign it, you are consenting to it. If you want to charge a lump sum for your services and can establish that you completed those services, then you should only have to provide your invoice to the company, along with an affidavit that you have paid for all labor and materials if they require that.
there is nothing illegal or improper about what this company is doing, although it may be a little tough on you. I think you face a business decision: can you get them to change their requirements; if not, are you willing to submit to their requirements; if not, are you willing to walk away from them? That should be your analysis. Good luck.