Kuznetsov's Sequence

PowerShell v2+, 89 bytes

param($n)for(){$r=-join"$n"["$n".length..0];if(!($n=(($r%$n),($n%$r))[$n-gt$r])){exit}$n}

Iterative solution. Lengthy because there's no easy way to reverse an array, so we stringify it and index on it backwards to store into $r. Then a pseudo-ternary to pull out the appropriate modulo and re-store into $n for the next round. However, if the result is zero, that means the !($n...) will be $true, so we exit instead of $n. The numbers are left on the pipeline and (implicitly) returned as an array, but without an encapsulating pipeline or saving the results into a variable, the default Write-Output sticks a newline between.

Try it online! (Yes, dead serious.)
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Perl, 43 38 + 1 = 39 bytes

Run with the -n flag

say while$_=($;=reverse)>$_?$;%$_:$_%$

Try it online! Includes the two non-empty examples.

Explanation chart

-n: Wraps the entire program in while(<>){ ... ;}. This turns the above code into the following line: while(<>){say while$_=($;=reverse)>$_?$;%$_:$_%$;}. Notice, a semicolon has been added to the trailing $, so it now becomes an instance of the variable $;. In the condition of a while loop, <> automatically reads one line of input and saves it in the $_ variable. So now let's look at what the interpreter reads inside the outer while loop:

say while$_=($;=reverse)>$_?$;%$_:$_%$;
[op][mod][         condition          ]     #While is acting as a statement modifier.
                                            #It evaluates the operation as long as the condition is truthy.
            ($;=reverse)>$_?$;%$_:$_%$;     #The meat of the program: a ternary operation
            ($;=reverse)                    #The reverse function takes $_ as a parameter by default, and reverses the value.
                                            #The value returned by reverse is stored in the variable $;
                        >$_                 #A condition asking if $% is greater than $_.  Condition of the ternary operation
                           ?$;%$_           #If true, then return $; modulo $_
                                 :$_%$;     #If false, return $_ modulo $;
         $_=                                #Assign the result of the ternary operation back into $_
                                            #If $_ is non-zero, then the condition is true, and while will evaluate the operation
say                                         #Implicitly takes the $_ variable as parameter, and outputs its contents

Original code, saved for posterity: 43 + 1 = 44 bytes

say$_=$%>$_?$%%$_:$_%$%while$_-($%=reverse)

Pyth, 13 12 bytes

t.u|%F_S,s_`

Thanks to @TheBikingViking.

Try it online: Demonstration

My old code:

W
W=Q%F_S,s_`

Try it online: Demonstration

Explanation:

t.u|%F_S,s_`NNNQ  implicit Ns and Q at the end
               Q  start with N = Q (Q = input number)
        ,         create a pair with the numbers
         s_`N        convert N to string -> reverse-> convert to int
             N       and N
       S          sort
      _           reverse
    %F            fold by modulo
   |          N   or N (if the result is zero use N instead to stop)
 .u               apply this ^ procedure until a value repeats
                  print all intermediate values
 t                except the first one (the original number)