Recent news articles related to this paper report the claim that Earth's solid inner core sometimes rotates backwards. The literal claim (as it appears in headlines) seems to make no sense given a basic understanding of friction and conservation of angular momentum. The matter is discussed in this SE question, but there is also the matter of multi-decadal cycles. What is the actual assertion? Is the inner core thought to be cycling between prograde and retrograde rotation, or is it something more subtle, like a wobble, sometimes slightly leading and sometimes slightly lagging, and at what amplitude? Is it a matter of "gaining" or "losing" a few degrees over some number of years (so would only be rotating "backwards" in the frame of reference of the crust, not non-rotating space)?
The mantle rotates about 131850 degrees per year. The actual assertion is that the inner core cycles between rotating about 131851 degrees per year versus 131849 degrees per year over the course of 70 year cycle. The paper was only published yesterday, so the scientific consensus is not there yet. The scientific consensus is not there yet on work done by the same authors eight years ago.
The inner core rotates in the same direction as do the mantle and crust, and rotates at almost at the same rate as the mantle and crust. The claim is that this rate varies by a tiny fraction compared to the mantle's rotation rate, plus or minus a degree or so per year compared to the mantle (but keep in mind that is one degree out of 131850 degrees). It does not switch directions.
David Hammen's excellent answer covers the key pieces, but the original question also has this part:
The literal claim (as it appears in headlines) seems to make no sense given a basic understanding of friction and conservation of angular momentum.
This isn't all that simple, actually. What is being conserved is the angular momentum of the Earth as a whole, not of the Earth core in isolation. As a consequence, the Earth's inner core can slow down a little bit if the rest of the Earth accelerates a little bit, and the other way around. Of course there needs to be a torque between the two, but that is not hard to fathom: We know that the outer core is moving rapidly (on the order of centimeters per second), due to thermal and/or chemical convection, and so it is not hard to imagine that there is friction between inner and outer core.
This is not so different from the fact that the solid Earth does not rotate at a fixed rate, but that days are fractions of seconds longer or shorter because the average east-west speed of the atmosphere changes from day to day; because the overall angular momentum must stay the same, an accelerating atmosphere implies a slowing down solid Earth, and the other way around. The same kind of thing happens (on slightly longer time scales) with ocean currents that take on more or less angular momentum and, as a consequence, have an influence on the rotation rate of the solid earth.