I have derived some properties regarding the Fourier transform and conjugation, I would like to make sure I have not made any mistakes and those make sense:
$$\int_{\mathbb{R}^d}f(x)\exp(-2\pi i \langle u,x \rangle)\,dx = \hat{f}(u)$$ $$\mathcal{F}[f^*](u) =\int_{\mathbb{R}^d}f^*(x)\exp(-2\pi i \langle u,x \rangle)\,dx = \overline{\int_{\mathbb{R}^d}f(x)\exp(2\pi i \langle u,x \rangle)\,dx} = \overline{\mathcal{F}^{-1}[f]}(u)$$ $$\int_{\mathbb{R}^d}\hat{f}(u)\exp(2\pi i \langle u,x \rangle)\,du = f(x)$$ $$\mathcal{F}^{-1}[\hat{f}^*](x) = \int_{\mathbb{R}^d}\hat{f}^*(u)\exp(2\pi i \langle u,x \rangle)\,du = \overline{\int_{\mathbb{R}^d}f(u)\exp(-2\pi i \langle u,x \rangle)\,du} = \overline{\mathcal{F}[\hat{f}]}(x)$$
So in some sense to swap the places of the Fourier transform and conjugation I need to invert the operator? Notably, if $f$ is real, then $f=f^*$, which implies that: $$\mathcal{F}[f](u) = \mathcal{F}^{-1}[f](u)$$
Which I believe corresponds to the fact that: $\hat{f}(-u) = \hat{f}^*(u)$ for real functions. Are there other properties related to this?