I know people who work at both companies, and they are both really good places to work but different. From what I've heard, Exxon probably isn't very Aspie-friendly. I've heard that the work environment there is extremely competitive, and that everyone is out to get each other in a sense. It's very focused on making money, and most of the employees are after just that. You can become extremely successful there, but be prepared to sacrifice a lot to do so.
Dow, on the other hand, would probably be an excellent place to work, Aspie or not. You may not make as much money as at Exxon, but money isn't everything. Everyone I've known who has worked for them has had extremely positive experiences, and says that the company in general promotes very respectful and safe work environments.
Both of these are coming from an outsider's perspective, but I'll wish you all the best!