I resigned and gave two weeks notice. The manager then told me to leave and not to work out the notice. When I begin working at the new company, should I tell them, if they ask, that I wasn't allowed to finish the term of my notice?
That kind of thing happens, and it is most likely due to the company, and not due to anything you have done. So that's nothing to worry about as far as your reputation is concerned.
More importantly you should check two things: One, are you being paid for the notice period. Second, are you on gardening leave (meaning you are still employed for two weeks, but you are neither allowed nor required to appear at work or do anything for the company), or have you received a payment in lieu of notice (which means you are now unemployed, but with 14 days pay in your pocket). The difference is that in the second case, you could start earlier with your new employer and make twice the money for two weeks.
If you don't get paid (which may be legal in some countries, but wouldn't be legal in most of Europe), then obviously you can ask the new company to start earlier in order to avoid in involuntary unpaid holiday.
I don't see why you should lie or hide the truth.
It is rather common to terminate the working relationship as soon as possible. You might reformulate it. It is not that you were not allowed to finish the term of the notice; it's more that you were released sooner.
Bear in mind you are providing a service and, by giving notice, your employer might decide the best service you could provide to the company is by not being there.
Ultimately, it's up to them because they are paying (you are entitled to be paid for this time, btw).
In general, trying to hide things are not worth it in the long term.
There is only 1 situation where it is relevant that you don't need to work during the notice period:
If you already mentioned that you need to give notice and would not be able to start work right away AND you want to start work right away.
In any other case, I would not mention it. Even if it is quite normal under some circumstances (from a company perspective to minimize risk/moral loss) it will never be a plus for you to mention this, unless it should prompt action.
Some practical answers:
- When are you available: Now
- When did you give notice: I recently gave notice and am now available
- You mentioned last week that you gave notice, would you like to start next week: Yes
You can say, 'oh it is actually possible for me to start now', but why complicate matters?
Sidenote: Even though you are not required to work anymore, you may not yet be relieved of your contractual obligations. In practice it should usually be fine, but in theory there could be something in your contract to prevent you from starting work before the other contract is actually finished.
I could imagine this to come up if you switch to a competitor who is trying to win the same bid this week.