Whether lifting weights and cycling are complimentary or not is a function of your goals. Lifting weights is about building strength and muscle fiber recruitment. Both are important for cycling. This month has been about racing, lifting weights to increase strength and hard training. Next month will be racing, hard training and strength maintenance which will include leg work, upper body work and core in the gym. I have only so much energy and recovery capacity so goals drive the amount and mix of training, strength work and racing. I am slower on the bike when I am working my legs harder in the gym and I feel it the next day.
As you can imagine, everyone has an opinion on whether weight training adds any value to cycling - some yes and some no. I like to lift weights and workout in a gym so it is a no brainer. I do it.