The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional twisted cube, denoted by $TQ_n$, an important variation of the hypercube, possesses some properties superior to the hypercube. Recently, some interesting properties of $TQ_n$ were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in $TQ_n$ for any odd integer $n\geqslant 5$. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing two algorithms that require a ring structure by allowing message traffic to be spread evenly across the twisted cube. Furthermore, we construct two equal node-disjoint cycles in $TQ_n$ for any odd integer $n\geqslant 3$, in which these two cycles contain the same number of nodes and every node appears in one cycle exactly once. In other words, we decompose a twisted cube into two components with the same size such that each component contains a Hamiltonian cycle.