Physical 2^4 Methods

Notation

Because this is a physical puzzle, we can easily adapt x, y, & z rotations to fit the moves. The puzzle is held horizontally throughout most of the solve, so the L and R cells can do any x, y, & z rotations freely. The I and O cells can do any x rotation, as well as only y2 or z2 rotations. The other sides have restricted turning due to the projection, and can only do 180 degree twists, so they will just be referred to as U2, D2, F2, B2.

Gyro Algorithm

Gyro algorithm

From a solved puzzle, there's nothing you can do to change the orbits of the L/R stickers, so we need an algorithm to do a special 4D rotation of the puzzle, called a gyro.
A commonly used algorithm for the Gyro is:

• Take the left endcap off and put it on the right so it becomes the right endcap (this brings the puzzle into the inverted state)
• Ly Ry'
• Take the right endcap off and put it on the left so it becomes the left endcap (this brings the puzzle back into the normal state)
• Rx2 B2 D2 Lx2

Note that the last 2 moves (D2 Lx2) could be replaced by D2 Rx2, U2 Lx2, or U2 Rx2 based on the solver's preference.
Watch [Melinda's 6 Snap Gyro] for an alternative algorithm.

Orienting Both Cells

Grant's Method

Exactly 8 oriented to the U/D axis

The first step is to get EXACTLY 8 pieces from an opposite colour group oriented to U/D. They can be in any position, as shown in the diagram, but must be oriented to the U/D axis. Make sure that there are exactly 8, then perform the gyro algorithm.
Tips:

• Use inspection time to count which of the 4 sets of colours has the most oriented to U/D, and start with that set.
• This should only take a few moves, and is very intuitive.

8 on L, 8 on the U/D axis of R

Next, pair up the other 8 pieces into a cell where they are oriented to U/D. This will end up separating the pieces that you oriented in the last step onto it's own cell.
From this position, you want to rotate the R cell so the pieces are oriented to the I/O axis (a z or z' rotation). Then perform the gyro algorithm, but undo the last 2 moves of the gyro. Then just undo that Rz or Rz', and do another gyro.
Now you will have all 16 corners oriented to the L/R axis.

Permuting Both Cells

CBC

Directly solve one of the cells, then use RKT to solve the other cell

RKT Parity

If you end the solve and a single layer needs twisting by 180 degrees, then do this algorithm with RKT:

• R2 B2 R2 U R2 B2 R2 U

There is also [this] video by Melinda with many alternative algorithms.