I'm so sorry this is happening to you. Although I am not licensed in your state, I would say that you should take a look at the contract you all signed. Most likely, it doesn't say anything about the ATO being paid. If you signed the contract, then, I would say that you don't really have a leg to stand on. If you need more assistance, you should consult with the Oregon Department of Labor.
Oregon is an at will employment state. There is no requirement to provide paid time off or vacation. That is a matter of contract. So it will depend specifically on the contract. If the contract clearly specifies that it is not paid, then you have little chance of prevailing. If it is vague and purposely hides the fact that it is not paid, then you may have a chance to prevail. If all nine of your team are able and willing testify that you were lead to believe that authorized time meant paid time, and further that authorized simply denoted that you needed to have permission regarding when tot take your time off, then you have a good shot. It really depends upon the specific facts, the contract, who will testify, and what the fact finder believes.
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A contract is interpreted by what is in the contract and the court is not allowed to add to or construe the contract unless there is an ambiguity. It may be the contract is confusing because of the way the two terms were used and if so, then I think you may have an opportunity to rescind (ie cancel) or reform (ie amend) the contract but before making that determination - which would require a full explanation of the facts and how it was presented as well as what the written document says - I would want to know if the reference to "the team" means you were brought here from somewhere else, for instance to work in the tech industry (which could raise legal issues), and what kind of value you bring to the company. Winning could take a long while -and if the industry is small you may develop a reputation as a trouble maker before it is all resolved - so the first question is whether you all are willing to ask to have that aspect of the contract renegotiated (maybe get half paid and half not) and be willing to leave if the company says no. If you are a valuable part of the company, it maybe willing to renegotiate but if you are not, or the team didn't perform, or the company is in financial trouble, you may not have many options. Cases like this are difficult - there are remedies under state and federal laws if the company treated you in ways inconsistent with its legal obligations (say, for instance, if it was openly discriminatory in specified ways) but a breach of contract in an at will state really gives you only the benefits of the contract - no real opportunity for enhanced damages and litigation takes time and energy.