Intuition for the homology of the klein bottle, specially the $\mathbb{Z}_2$ part

Question

I understand the method of getting the homology but I dont know how to interpret it: Ive always interpreted 1-homology as the number of independent 1-cycles you can do on a surface. For example for the torus with homology $H_1(T)=\mathbb{Z \times{Z}}$ means you can have two independent 1-cycles that you can combine infinitely with each other. But with the klein bottle this intuition doesnt seem to work as the 1-homology is $H_1(K)=\mathbb{Z_2 \times{Z}}$, what does the $\mathbb{Z_2}$ term means? Does combining 2 times the 1st independent path is equivalent to not doing any path at all? How is that possible?