Logic Level Translator Operation
One characteristic of MOSFETs is that the drain pin can act as a source (and vice versa) depending upon the voltages applied to the terminals.  That is exploited in this circuit.

When there is a low level on the high-voltage port the body will conduct diode and pull down the voltage on the low-voltage port until the voltage between the gate and the drain pin exceeds the threshold voltage of the device.  The drain pin acts as a source pin in this case. The positive bias causes the MOSFET to conduct from source to drain and pull down the voltage on the low-voltage port down to a very low voltage (only millivolts above the voltage on the high-voltage port.  I measured a BSS138 and discovered that the threshold voltage in this inverse mode is very similar to that in normal mode - I couldn't find it on any data sheets.

The method of fabricating MOSFETs creates the diode (referred to as a body-diode) as a side-effect - in this case the body diode does not affect operation significantly except for the case where the voltage on the high-voltage port is low enough to cause the body diode to conduct but there is not enough voltage difference between the gate and the drain pin to turn the device on - in this case the low-voltage port will be at a voltage abut 600mV above that on the other port.

As commented by @Michael Karas the threshold voltage needs to be significantly lower than the low voltage source to operate correctly.

Also as commented by @Michael Kara the physical construction of discrete devices is not symmetric but for basic analysis of the circuit it can be treated as such with the body diode in parallel.