It's difficult to provide a meaningful analysis based on the facts set forth in your post. To enforce your rights, you have probably figured out that you will need to consult with an attorney one-on-one. There are several law firms in your area that can assist you with this matter.
Generally speaking, if you are true independent contractor you must incur those costs yourself and then you will be entitled to write them off on your (sounds like) Schedule C. If the client/business is giving you the reimbursement, then they can properly issue a 1099 C for the nonemployee compensation. You then write it down (as you otherwise would be entitled) as an expense of running your business. My concern here, as I am sure others are thinking, is that you may eventually have a problem with actually being an independent contractor. There is a 12 point test the IRS uses to make that determination, and there are not enough facts here to make that call. See a tax attorney in NOVA.
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It makes no difference to you from a tax perspective how they report it. If you are self employed (which you must be if you are an independent contractor, you have to report the reimbursement as income (because it is). Than income is offset by the deductions you are taking for these business expenses.