In this paper, we investigate the properties of spreading speeds of the following Fisher-KPP equation in almost periodic media: \(\begin{aligned} \left\{ \begin{aligned} u_t(t,x)=\mathcal {M}u(t,x)+f(x,u(t,x)),\ t>0, x\in \mathbb {R},\\ u(0,x)\ge 0,\ u(0,\cdot )\ne 0\ \text {with compact support,}\\ \end{aligned} \right. \end{aligned}\) where either \(\mathcal {M}u(t,x)=\mathcal {M}^ru(t,x):=\partial _x(a(x)\partial _{x}u(t,x))+b(x)\partial _{x}u(t,x)\) , which represents the random dispersal, or \(\mathcal {M}u(t,x)=\mathcal {M}^nu(t,x):=\int _{\mathbb {R}}\big (u(t,x-y)-u(t,x)\big )d\mu (y)\) , which represents the nonlocal dispersal. With the existence of the spreading speeds \(\omega ^\pm \) in the positive and negative directions of the equation at hand, we 1. give a sufficient and necessary condition for \(\omega ^+=\omega ^-\) , which means that the propagation of the solution is symmetric when \(\mathcal {M}=\mathcal {M}^r\) ; 2. illustrate that the condition above is a sufficient but not necessary one when \(\mathcal {M}=\mathcal {M}^n\) ; 3. give some other sufficient conditions for \(\omega ^+=\omega ^-\) when \(\mathcal {M}=\mathcal {M}^n.\)
