Googling gives me the answer to use Font::TTF::Name but the docs are hard to figure out. However I did get some results with the script ttfwidth.pl from the Examples directory of Font-TTF-Scripts, which gave output like

$ ./ttfwidth.pl -u Symbola605.ttf  

There are 6796 glyphs
font mapping Microsoft id = 3, encoding = 1
    (encoding => UGL coding)
Unicode, Glyph, AdvWidth, LSdBearing, Xmin, Xmax, Ymin, Ymax, XCentre
0x0020,3,512,0,0,0,0,0,512
0x0021,10,682,229,229,453,0,1466,341
0x0022,11,809,108,108,701,771,1339,404
0x0023,12,1024,39,39,985,-121,1265,512
0x0024,13,964,108,108,854,-108,1446,483
0x0025,14,1662,50,50,1612,-134,1578,831
0x0026,15,1593,95,95,1498,-45,1466,796
...etc
...etc
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and I did have some good results with

#!/usr/bin/perl
use warnings;
use strict;
use Font::TTF::Font;

my $font = shift || "./Symbola605.ttf"; 

my $f=Font::TTF::Font->open($font) or die "$!\n";; 

my @out = @{$f->{'post'}->read->{'VAL'}};
#print "@out\n";
my @unihex =  grep{/^u[\d+]/} @out;
print "@unihex\n";
[download]

But my head is starting to spin now, and I think I will set it aside for awhile, until unicode support gets better. Even the app gucharmap fails to display all the glyphs, even though my Tk program can.