Let $X$ be a spectrum. Is there a canonical construction/functor that would associate to this spectrum, an inverse spectrum $X'$, in the sense that $$\pi\_\*(X)\cong \pi\_{-\*}(X')?$$


To be more precise, such a spectrum $X'$ can always be constructed by attaching cells to produce the right homotopy groups, but is there a more conceptual way of creating it?



Inverting homotopy groups of spectra
Not exactly what you asked for, but there is the Brown-Comenetz dual of $X$, $I\_{\mathbb{Q}/\mathbb{Z}}X$, which has the property that its homotopy groups are Pontryagin dual to the homotopy groups of $X$ (in the negative degree):


$\pi\_{-\ast}(I\_{\mathbb{Q}/\mathbb{Z}}X) \simeq Hom(\pi\_\ast(X),\mathbb{Q}/\mathbb{Z})$


So, if the homotopy groups of $X$ are all finite, then the Pontragin dual groups will be (non-canonically) isomorphic and this will have the required property. 


[See here for more details](https://ncatlab.org/nlab/show/Brown-Comenetz+duality)


