How is symmetry in an inequality determined?

I was reading a book about inequalities, in that I found that

$$\begin{array}{}\text{(1)}& \frac{a}{b}+\frac{b}{c}+\frac{c}{a}>1\end{array}$$

is a symmetric inequality in a,b,c. But if I change the order from (a,b,c) to (a,c,b), I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?

I was reading a book about inequalities, in that I found that

$$\begin{array}{}\text{(1)}& \frac{a}{b}+\frac{b}{c}+\frac{c}{a}>1\end{array}$$

is a symmetric inequality in a,b,c. But if I change the order from (a,b,c) to (a,c,b), I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?