Let's start with determining velocity $\newcommand{\v}{\mathbf{v}}\v$ from angular velocity $\newcommand{\w}{\boldsymbol{\omega}}{\w}$. If an object is currently at position $\newcommand{\r}{\mathbf{r}}\mathbf{r}$, and is rotating about a fixed point, which we will take to be the origin, with angular velocity $\w$, then the object's velocity is given by $\v =\w \times \r$.
Now to find the object's linear acceleration $\newcommand{\a}{\mathbf{a}}\a$, simply differentiate the above equation:
$\begin{equation}
\begin{aligned}
\a = \dot{\v} &= \dot{\w}\times \r + \w \times \dot{\r} \\
&= \newcommand{\al}{\boldsymbol{\alpha}}\al \times \r + \w \times \left( \w \times \r \right) \\
&= \al \times \r + \w \left(\w \cdot \r\right) - \omega^2 \r \\
&=\al \times \r - \omega^2\left(\mathbb{I} - \hat{\omega}\otimes\hat{\omega} \right)\r.
\end{aligned}
\end{equation}$
Above, the second line introduces the angular acceleration $\al$, defined as the time derivative of $\w$. Also in the second line, but in the second term, we used the result for velocity that $\dot{\r} = \v = \w \times \r$. In the end, we got that the linear acceleration $\a$ consists of two terms. The second term is the usual centripetal acceleration term, which looks like $-\omega^2r$, but there is a projection which makes sure you are using the separation from the closest point on the axis of rotation (that is, you subtract of the component of $\r$ along $\w$).
The tangential acceleration, then, must be contained in the first term. Since it is given by a cross-product with $\r$, we see it is perpendicular to $\r$ and therefore is "tangential" in the sense that is tangent to the sphere of radius $r$. Notice in the special case where the axis of rotation is fixed, so that $\a$ and $\w$ are colinear, $\a= \al \times \r$ is colinear with $\v = \w \times \r$, so the tangential acceleration is colinear with the velocity as expected.
Best Answer
Torque, like angular momentum is a pseudovector and not a vector. It is a conventional way of showing the direction (anti/clock -wise) ot the rotation. It is devised in such a way that you can apply the right-hand rule.