Osculating varieties and their joins: $\mathbb{P}^1\times \mathbb{P}^1$

Let $X\subset \PP^r$ be an integral projective variety. We study the dimensions of the joins of several copies of the osculating varieties $J(X,m)$ of $X$. Our methods are general, but we give a full description in all cases only if $X$ is a linearly normal embedding of $\PP^1\times \PP^1$. For these embeddings of $\PP^1\times \PP^1$ we give several examples and then study the joins of one copy of $J(X,m)$ and an arbitrary number of copies of $X$.