What could be causing this strange behavior of a branch circuit?
Sounds like there's a break in the neutral wire between the panel and the first receptacle, or possibly a bad neutral connection at the panel. 

Power is going out on the ungrounded (hot) conductor, passing through a load somewhere on the circuit and returning on the grounded (neutral). However, since the neutral is not continuous back to the panel, the circuit is not complete.  When you put the lamp between the grounded (neutral) conductor and the grounding conductor, you completed the circuit through the lamp. 

If you measure with a digital multimeter, you should measure 0 volts between the ungrounded (hot) conductor and the grounded (neutral) conductor. You'll measure a voltage (1 - 125 volts) between the grounding conductor, and either the grounded or ungrounded conductor. 

If you disconnect the grounded (neutral) conductor from the receptacle,  and measure continuity between it and the grounding conductor.  You'll get an open (or infinite) reading,  because the neutral is not continuous back to the panel.  

Fix the break or loose connection, and your problem will be solved.