Can a LDL decomposition be efficiently converted into UDU decomposition?

Question

Question: If you are given matrices $L,D_L$ where $L$ is a lower unit triangular matrix, $D_L$ is a diagonal matrix, and $L,D_L$ are a LDL decomposition of a particular symmetric positive definite matrix $A$ as, $$A=LD_LL^T$$ can you calculate matrices $U,D_U$ where $U$ is an upper unit triangular matrix, $D_U$ is a diagonal matrix, and satisfices $A=UD_UU^T$?
Of course, there is a obvious solution by calculating $A$ from $L,D_L$ and then conducting a Cholesky decomposition to acquire $U,D_U$, but can this be done without explicitly calculating $A$?
Notes: I found a relation between LDL decomposition and UDU decomposition as $$UD_UU=(L'^{T})^{-1}(1/D')(L')^{-1}$$ $$A^{-1}=L'D'L'^T$$ which shows that $U,D_U$ can be calculated by solving the LDL decomposition of $A^{-1}$, but I don't know if this is a path to a solution.