Have you tried swapping the wires? Swapping them should do the trick. You only have two wires to chose from on these signals, correct?
Side note of info... DC voltage really matters in which orientation of your wire leads. Direct Current (DC) only has one path and no frequency which is opposite of Alternating Current (AC) which travels both negatively and positively and the frequency is the rate of which it travels; ie. 60 Hz (frequency), there are 60 pulses within one second.
When at home you can hook up AC either way with your wire leads and it doesn't matter due to the current traveling both positive and negative on your sine waves. On generators and rectifiers, diodes (diode packs more specifically) are used to convert AC into DC by orientation of the diodes to allow the current to travel only one way and not the other and the result is DC voltage/current.
The fact that there is no flashing when you have it wired right indicates that the relay is bad.
The wiring diagram at the web site for the relay shows the positive lead (Brown Wire) goes to the X terminal #49 on the relay and the power out (Orange Wire) goes to the L terminal #49a.
If you have it wired that way and it doesn't work, the relay is most likely defective.
Both sides light up because the dash indicator bulb cross feeds between the right and left sides so the single indicator bulb can work for both sides. In stock form the indicator bulb is so small compared to the signals that it does not cross feed enough current to light the opposite side, but with LED signals the incandescent dash bulb cross feeds enough current to light them.
From Tom Schmitz's comments it appears that the small #194 LED dash bulb he is using cross feeds so little current that the opposite side don't light up. That LED #196 indicator bulb is and has to be non-polarized so that it can cross feed to both sides or he has wired in some diodes to solve the problem.
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