Let \(X\subset \mathbb{P}^r\) 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 \(\mathbb{P}^1\times \mathbb{P}^1\). For these embeddings of \(\mathbb{P}^1\times \mathbb{P}^1\) we give several examples and then study the joins of one copy of \(J(X,m)\) and an arbitrary number of copies of \(X\).