How many lives would be saved if the airlines turned all the seats around?
The evidence is not the most recent one, but


  According to an article in the December 1952 edition of Naval Aviation News, “Passengers in Navy transport planes have ten-fold better chances of coming out of crashes alive, thanks to backward-facing seats which are being installed in all new planes […]". The unsigned article cites two Royal Air Force accidents involving a four-engine Hastings and a two-engine Valetta. Both had rear-facing seats that were credited with minimizing injuries to passengers.


Unfortunately, the link to the 1952 edition ends in a 404. Thanks to the detective work by Gerald Schneider we now have a working link. The article is on page 27. Another source cites a 1957 study, without a traceable source:


  An Air Force study in 1957 reviewed a series of crashes and concluded
  that injuries were seven times greater among passengers facing
  forward.


That a backward-facing seat is safer is indisputable, but quantitative data is very hard to find. Given the much higher wing loadings and landing speeds of modern airliners, possibly those ratios cannot be translated to modern aviation. While in the days of propeller-driven transports the difference between the forward- and backward-facing position might indeed have meant that the latter provided a significant advantage in survivability, the deceleration level in a modern airliner crash might make both equally unsurvivable. In addition, the data from mostly young, fit and healthy subjects involved in crashes of military transports might not translate easily to todays untrained, middle-aged and overweight passengers.

Circumstantial evidence cited in this 2016 study, however, indicates that even for modern airliners seating position is important. For an A340 runway overrun accident in Toronto in August 2005 it cites the Canadian TSB as saying:


  ‘One of the cabin crew, seated in the same general area as the crew
  and passengers who incurred serious impact injuries, was not injured.
  This cabin crew’s seat was aft-facing; the other seats were
  forward-facing.’


Another more recent paper gives more modest advantages: This 1996 study by the European Transport Safety Council lists them in a table and awards rearward-facing seats with only a 5% advantage overall or 19% if based only on impact fatalities:



Commenting on the table, the numbers are based on the 1995 Cherry study (CHERRY, R.G.W. and Associates Ltd (1995) Analysis of factors influencing the survivability of passengers in aircraft accidents Vol. I-III. EU DG VII contract) and 


  These particular figures are not quoted in the study, but have been
  calculated from the data provided in the study report.