Another method that reads the data in paragraph mode, splits into first element and remaining elements within lines in one map then swaps the first elements of the two lines and rejoins things up in a second, finally joining the pairs of lines into paragraphs again.

$ perl -Mstrict -Mwarnings -E '
> open my $fh, q{<}, \ <<EOF or die $!;
> jay 21 34 56
> bob 12 39
> 
> jay 11 10
> bob 14
> 
> jay 190
> bob 13
> EOF
> 
> say join qq{\n\n}, do {
>    local $/ = q{};
>    map {
>        ( $_->[ 0 ]->[ 0 ], $_->[ 1 ]->[ 0 ] )
>        =
>        ( $_->[ 1 ]->[ 0 ], $_->[ 0 ]->[ 0 ] );
>        join qq{\n}, map { join q{ }, @$_ } @$_;
>    }
>    map {
>        chomp;
>        [ map { [ split m{\s}, $_, 2 ] } split m{\n} ];
>    }
>    <$fh>;
>    };'
bob 21 34 56
jay 12 39

bob 11 10
jay 14

bob 190
jay 13
$
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I hope this is of interest.