I am travelling by car in the USA (Massachusetts). Am I supposed to respect traffic signs that are upside down?
E.g. on this road, there have been an upside down 25 MPH speed limit for a while, I have no idea whether I am allowed ignore it:
Yes, any sign posted in the US is intended to be followed unless it explicitly has an X over it or it is covered up. Most likely the top screw of this sign has come loose and no maintenance has been done to fix it. Typically signs that are meant to not be respected are covered with black plastic or a board so they aren't even legible.
However, yellow signs like the one in the picture are advisory and following them exactly is usually not obligatory.
That is definitely still a valid sign, although a lawyer could probably use the fact that the sign was damaged to avoid a related traffic ticket.
It's also a yellow warning sign, which means that the speed limit is still that which was posted on a white sign. You cannot be ticketed for going the speed limit, although the implication is that if you do hit a bicycle or a pedestrian in a crosswalk, you'll be more liable for negligence than if you had slowed down. Basically, if it's a time of day when foot traffic is high in the area, you probably shouldn't be doing 35. But if it's 4am and you're the only one on the road, you definitely don't have to slow down.