This answer was based on a prior version of the question
I'm going to convert my comment to an answer and elaborate a bit on a comment left by @Tester101.
What you're describing here is not possible with residential electrical, unless you are testing with the washer plugged in (in which case all bets are off).
A 240V reading across your hot leads demonstrates that they are properly connected to the opposite phases of your supply. There is no other way reasonable way you could get such a reading.
The fact that one hot lead to neutral reads 120V likewise indicates it is correct. The voltage of a single supply phase to the neutral is 120V.
The wildcard (and user error, in my opinion) is that you claim the other hot lead reads 240V to neutral. The supply cannot be wired this way, and no erroneous circuity I can think of would result in such a reading.
For simplicity, you expect one hot leg to be -120V and the other to be +120V. Then, with neutral of zero you get 240V between the hot legs and 120V between each and the neutral.
Potential is usually considered from the RMS of the AC power wave. And this reading is compared additively.
The only answer applicable here is that your multi-meter is malfunctioning or you are using it improperly.
Here is the relevant math for why your description cannot be accurate (I did it on paper, easier than using the SE math writer, imho)
Section at the top-left are your readings.
Jackie Chan was probably the more appropriate drawing, but I don't have those skills....