Heres a slightly dodgy bash only script, if you don't want to/can't install BioAwk/Perl/Python etc. It assumes the input files are perfectly formatted fasta's with headers as per the OP, i.e. >xxxx|yyyyy|zzzzzzzzzz

(I haven't tested it with fasta's that weren't already linearised but it should work...)

#!/bin/bash
# $ script.sh sequences.fasta

while read line ; do
    if [ ${line:0:1} == ">" ] ; then
        desc1=$(echo $line | cut -d '|' -f 1)
        desc2=$(echo $line | cut -d '|' -f 2)
        desc3=$(echo $line | cut -d '|' -f 3)
    else
        seq=$line
    echo -e "$desc1|$desc3|$desc2"
    echo -e "$seq"
    fi
done < $1

In this script, the sequence fasta is looped over line by line, if it hits a header line (i.e. starts with a ">"), which in bash is expressed here as ${line:0:1} (the zero-th to first character range, AKA, the first character, is a ">"), the string is split on the | (pipe symbol) delimiter, and cut then picks out each section of the header and assigns it to a variable.

If the string didn't match the ">", it must be a sequence line, and can just be re-emitted as it is.

echo then prints each variable, separated by | again, in the order you want (1, 2, 3 becomes 1, 3, 2).

EDIT: a slightly neater version

#!/bin/bash
# $ script.sh sequences.fasta

while read line ; do
    if [ ${line:0:1} == ">" ] ; then
        IFS='|' read -a header <<< "$line"
    else
        seq="$line"
    echo -e "${header[0]}"\|"${header[2]}"\|"${header[1]}""\n""$seq"
    fi
done < $1

Same process basically, except this time we can make use of the inbuilt "IFS" (internal field separator), and tell it to use | instead. In conjuction with read -a we can turn a string, in this case the header line, in to an array, saving us a few lines of code.

We also don't have to handle multiple variable names explicity, since bash array syntax lets you address individual items with "${array[1]}" for example (which is zero based, so is actually the second element). "${array[@]}" would refer to every element.

If we include a "\n", we can also emit the fasta in a single echo line, though I personally would argue this is less neat (certainly less readable).