Is one allowed to participate in a Secret S-nta game? There is one happening at my place of work, and it will be very uncomfortable to be the only person not participating.
The Remo and Shach (very end of YD 148) discuss sending presents to a gentile on a religious holiday, and they rule (based on Terumas Hadeshen 195) that there is room for leniency if there is a concern of real animosity. (It is important to distinguish between animosity and mere discomfort.)
This Hetter is based on the presence of number of factors, namely – a) the concern of animosity, b) that gentiles these days are less devout, and they celebrate more for their own enjoyment than out of a sense of truly religious fervor or duty, and c) the gifted item is not a religious article, nor is it being used for religious rituals.
Even so, the above Poskim rule that it is best not to give the present on the day itself, but at least a day before (or a day after), to try as much as possible to disassociate the present from the religious holiday. The Remo also says, earlier in the same Seif, that it is appropriate for a Baal Nefesh to find ways around it entirely, where possible.
The standard text of the Shulchan Aruch, which was subject to government censorship, does not identify the specific religious holiday it was referring to. However, the uncensored edition states the above in relation to “the eighth day after Nittel, which they call New Year”. This is also clear in the Terumas Hedeshen, the source of the Din cited in Shulchan Oruch. In light of this, some wish to argue that the above leniencies apply only to New Year, and not Nittel, which is stricter. However, this assertion is far from definite; it could very well be that New Year was singled out simply because, historically, that is the time when presents were gifted. [This is very much implied in Terumas Hadeshen, and also seems supported by our knowledge of the historical facts.]
In addition to the concern of idolatry, it should be noted that this question also raises the issue of Chukos Hagoyim; the prohibition against adopting the practices of the nations. This issue is not explicitly discussed by the above Poskim in this context, but however we might explain it, the fact remains that this did not prevent them from granting the above leniency. [One possible explanation: Perhaps they concur with Maharik 88 that the prohibition of Chukos Hagoyim applies only to customs that are irrational or immodest.]
A: The best solution would be to avoid participating entirely in the present exchange.
B: If you absolutely need to participate in the present exchange, write an appreciative note in which any reference to the holiday, end of year, or any religious aspect is completely and conspicuously absent. Rather, speak about the present being for the great work relationship etc. As much as possible, it should be clear that it has nothing to do with the Yom Eid or any religious aspect. When you receive a present, discard it. Also, if the gift exchange takes place at a holiday-themed party, make sure to absent yourself from the ceremony.