A
zener diode behaves exactly the same as an ordinary diode
except the reverse voltage rating of a zener diode is the zener voltage. The reverse voltage rating of ordinary diodes can be just about anything, but heavy doping of the PN junction produces a specific reverse breakdown voltage for zener diodes. That transition from non-conduction to conduction is typically "sharper" for a zener diode, occurring at a well-defined and stable voltage. The conductance after crossing the zener threshold voltage is typically greater too. In any case, some means of current limiting is required when the applied reverse voltage exceeds the zener voltage. Unlike ordinary diodes, which are not intended for conduction in the reverse breakdown region, zener diodes are
designed to work in this region.
Your circuit does not provide for a negative voltage to be applied to the adjustment terminal via a zener diode, as was done with an auxiliary diode rectifier, capacitor, and current limiting resistor applied to the zener in the original Nuts and Volts schematic. Your zener is NOT installed up-side down, but its cathode needs a negative voltage applied with respect to circuit common. The zener voltage will appear across the
reverse biased zener diode, and since it is negative with respect to circuit common, that voltage will be applied to the adjustment terminal of the LM317 when R4 and R5 are adjusted to nearly zero resistance.
Unfortunately, there is no simple way to obtain the negative voltage you need from an untapped secondary winding connected to a bridge rectifier because you are already using the "negative" output of the bridge as your circuit common! To get a negative supply for the zener will require a small transformer (a 6.3 VAC filament transformer works fine), a rectifier, a filter capacitor, and a current-limiting resistor for the zener. Sorry 'bout that, but that's the way the current flows.