I want to understand if I can keep the x axis automatically "centred" around 0, i.e., having the absolute axis option values for x min and x max equal, without setting them explicitly.

The following MWE shows the issue. I have point estimates and their confidence intervals. The maximum values are -3.5 and 3.5, however, setting those to x min or x max would decrease the readability (shrink the plots).
On the other hand, comparing the point estimates would be easier if the zero was centred.

MWE:

documentclassarticle

usepackagepgfplots

beginfilecontentsdata.csv
nr est CL CU
1 -0.9 -1.3 -0.3
2 0 -0.6 0.58
endfilecontents

begindocument

begintikzpicture
beginaxis[xmajorgrids]
addplot+[only marks,error bars/.cd,x explicit,x dir=both] table [
 x=est, 
 y = nr,
 x error plus expr=thisrowCU-thisrowest,
 x error minus expr=thisrowest-thisrowCL]data.csv;
endaxis
endtikzpicture

enddocument

I am not sure if there is an automatic solution since pgfkeysvalueof/pgfplots/xmin and pgfkeysvalueof/pgfplots/xmax won't be set but it is easy to infer the actual values of xmin and xmax using calc. They can be used in the final plot.

documentclassarticle
usepackagefilecontents
beginfilecontents*data.csv
nr est CL CU
1 -0.9 -1.3 -0.3
2 0 -0.6 0.58
endfilecontents*
usepackagepgfplots
usetikzlibrarycalc
pgfplotssetcompat=1.16

begindocument
Step 1: measure the $x_mathrmmin$ and $x_mathrmmax$ used in the plot.

begintikzpicture
beginaxis[xmajorgrids]
addplot+[only marks,error bars/.cd,x explicit,x dir=both] table [
 x=est, 
 y = nr,
 x error plus expr=thisrowCU-thisrowest,
 x error minus expr=thisrowest-thisrowCL]data.csv;
 path (0,0) coordinate (O) (1,0) coordinate(X);
endaxis
path let p1=($(current axis.west)-(O)$),p2=($(current axis.east)-(O)$),
p3=($(X)-(O)$) in
pgfextrapgfmathsetmacromyxminx1/x3%
pgfmathsetmacromyxmaxx2/x3%
messagexmin=myxmin, xmax=myxmax^^J%
pgfmathsetmacronewxmin-1*max(abs(myxmin),abs(myxmax))%
xdefnewxminnewxmin%
pgfmathsetmacronewxmaxmax(abs(myxmin),abs(myxmax))%
xdefnewxmaxnewxmax;
endtikzpicture

Step 2: measure the $x_mathrmmin$ and $x_mathrmmax$ used in the plot.


begintikzpicture
beginaxis[xmajorgrids,xmin=-1.488,xmax=1.488]
addplot+[only marks,error bars/.cd,x explicit,x dir=both] table [
 x=est, 
 y = nr,
 x error plus expr=thisrowCU-thisrowest,
 x error minus expr=thisrowest-thisrowCL]data.csv;
endaxis
endtikzpicture

clearpage
You can also use the computed values.

begintikzpicture
beginaxis[xmajorgrids,xmin=newxmin,xmax=newxmax]
addplot+[only marks,error bars/.cd,x explicit,x dir=both] table [
 x=est, 
 y = nr,
 x error plus expr=thisrowCU-thisrowest,
 x error minus expr=thisrowest-thisrowCL]data.csv;
endaxis
endtikzpicture

enddocument

You may make it fully automatic by writing the inferred values to the aux file. I will be happy to spell this out if needed.