linear map transforming an orthogonal basis into an orthogonal basis

Question

Let $E$ and $F$ be two inner product spaces with the same finite dimension $n$.

Let T be an invertible linear map from $E$ to $F$.

let define the property :

H(T) : there exists an orthogonal basis $B_1$ in $E$ such that $T(B_1)$ is an orthogonal set of vectors of F.

Is H(T) always true if T is invertible ?

we can see that $ T \text{ is orthogonal} \Rightarrow H(T) $ but $H(T) \nRightarrow T \text{ is orthogonal } $

So the property H would be more general than orthogonality.