Como posso ordenar as linhas selecionadas alternativamente

Por favor, tente esta consulta abaixo. Usando ROW_NUMBER() OVER(PARTITION BY ..) gerar um número de classificação/registro para cada 'which' itens e classificar com base nisso. (Espero que isso funcione para você, eu não tenho o esquema da tabela ou o script de dados de amostra para tentar sozinho)

SELECT
  pr1.id AS user_id,
  pr1.title AS user_name,
  pr2.id AS liker_id,
  pr2.title AS liker_name,
  x.which AS which_table,
  x.cnt AS total
FROM 
(
  SELECT rid, rootid, which, COUNT(*) AS cnt
        ,ROW_NUMBER() OVER(PARTITION  BY which ORDER BY rid) AS new_order
  FROM
  (
    SELECT rid, rootid, 'vote' which FROM p_likes
    UNION ALL 
    SELECT rid, rootid, 'comment' which FROM p_comments
    UNION ALL 
    SELECT rid, rootid, 'friend' which FROM relations
  ) y
  WHERE y.rootid = 1246 AND y.rootid <> y.rid
  GROUP BY y.rid, y.rootid, y.which
) x
INNER JOIN pagesroot pr1 on x.rootid = pr1.id
INNER JOIN pagesroot pr2 on x.rid = pr2.id
ORDER BY new_order,x.cnt desc;