I don't recognize the notation used for the $x_i$ directions below (passage from textbook).
The components of a vector depend on the base vectors used to describe the components. This will also be true for tensors.
Let $\hat{e_1}, \hat{e_2}, \hat{e_3}$ be unit vectors in the direction of $x_{1^-}, x_{2^-}, x_{3^-}$, respectively, of a rectangular Cartesian coordinate system. Under the transformation $\boldsymbol{T}$, these vectors $\hat{e_1}, \hat{e_2}, \hat{e_3}$ become $\boldsymbol{T}\hat{e_1}, \boldsymbol{T}\hat{e_2}, \boldsymbol{T}\hat{e_3}$. Each of these can be written...
The book goes on without mentioning the notation anywhere nearby. And this is the first mention in the book.

He writes phrases like, "... about the $x_3$-axis... " quite a bit I'm seeing. So, in context, I'm gathering he mean axis by the dash symbol. That is, it's not a minus sign.