Perhaps you should just try it?

> { echo out; echo err >&2; } >f1 2>&1
> cat f1
out
err
> { echo out; echo err >&2; } 2>&1 >f1
err
> cat f1
out
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Redirection order definitely DOES matter in the shell. (Hmm, unless it's shell-specific? I'm using bash)

And just to do exactly what you said..

> perl -e'print "print\n"; warn "warn"' >f1 2>&1
> cat f1
warn at -e line 1.
print
> perl -e'print "print\n"; warn "warn"' 2>&1 >f1
warn at -e line 1.
> cat f1
print
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In the first case both print and warn go to the file, in the second case warn goes to stdout and print to the file.

~/stuff > perl foobar.pl 
Normal STDOUT
Going to STDERR
~/stuff > echo 'Just a starting point
> ' >JGBtest.txt
~/stuff > cat JGBtest.txt 
Just a starting point

~/stuff > perl foobar.pl >>JGBtest.txt 2>&1
~/stuff > cat JGBtest.txt 
Just a starting point

Going to STDERR
Normal STDOUT
~/stuff > cat foobar.pl 
#!/usr/bin/perl

use warnings;
use strict;

print "Normal STDOUT\n";
warn "Going to STDERR\n";

exit;

__END__
~/stuff >
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You must always remember that the primary goal is to drain the swamp even when you are hip-deep in alligators.

You didn't run the test with reversed redirection, so the above doesn't actually show anything new. Here's what I get with your code, including also a reversed order test:

~> perl foobar.pl
Normal STDOUT
Going to STDERR
~> echo 'Just a starting point
> > ' >JGBtest.txt
~> cat JGBtest.txt
Just a starting point
>
~> perl foobar.pl >>JGBtest.txt 2>&1
~> cat JGBtest.txt
Just a starting point
>
Going to STDERR
Normal STDOUT
~> perl foobar.pl 2>&1 >>JGBtest.txt
Going to STDERR
~> cat foobar.pl
#!/usr/bin/perl

use warnings;
use strict;

print "Normal STDOUT\n";
warn "Going to STDERR\n";

exit;

__END__

~>
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Which again shows that the redirection order does matter, and in the 2>&1 >>file case, STDERR goes to the output, not the file.

Update: And just to illustrate it in one more way...

~> perl foobar.pl >>JGBtest.txt  2>&1  &
[1] 22571
~> ll /proc/22571/fd
total 0
lrwx------ 1 user group 64 2014-10-29 16:31 0 -> /dev/pts/11
l-wx------ 1 user group 64 2014-10-29 16:31 1 -> /home/user/JGBtest.tx
+t
l-wx------ 1 user group 64 2014-10-29 16:31 2 -> /home/user/JGBtest.tx
+t
~> perl foobar.pl 2>&1 >>JGBtest.txt &
[2] 22578
~> ll /proc/22578/fd
total 0
lrwx------ 1 user group 64 2014-10-29 16:32 0 -> /dev/pts/11
l-wx------ 1 user group 64 2014-10-29 16:32 1 -> /home/user/JGBtest.tx
+t
lrwx------ 1 user group 64 2014-10-29 16:32 2 -> /dev/pts/11
~> cat foobar.pl
#!/usr/bin/perl
sleep(100);
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