One way is through a bilogarithmic plot.

Define

 bilog[val_, cut_: 1.] := Module[
 {out},
 out = If[Abs[val] <= cut,
 val,
 Sign[val] Log10[Abs[val]]
 ]
 ];
for the data and 

 

 blvs[{rl_, rh_}, cut_: 1] := Module[
 {out, lin, lgn, lgp, lgt, lgm, lgo, tik, tkn, tkp},
 lin = Range[-.9 cut, .9 cut, cut/10];
 lgp = Range[Log10[cut], Log10[rh], 1];
 lgn = Range[Sign[rl] Log10[Abs@rl], Log10[cut], 1];
 lgm = {#, Sign[#] 10^(Abs@#)} & /@ Join[lgn, lgp];
 lgo = Log10[Range[1, 9, 1.]];
 tkn = Sort@-(lgo + # & /@ Sort@Abs@lgn);
 tkp = Sort@+(lgo + # & /@ Sort@Abs@lgp);
 tkn = {#, ""} & /@ Flatten@tkn;
 tkp = {#, ""} & /@ Flatten@tkp;
 tik = Join[tkn, tkp, lgm]
 ]

for the frame ticks, then

 blv = {N@bilog[#[[1]], 1.], #[[2]]} & /@ llvaluefull;

and

 ListPlot[blv, 
 FrameTicks -> {{Automatic, None}, {blvs[{-10000, 10000}, 1], None}}]

gives you the plot I think you're after. Sans labels.

Another popular (at least in the geophysics community) transformation is through ArcSign, but I'll leave definition of frame ticks as an exercise for the reader.