The Clausenian hypergeometric function 3F23F2 with unit argument and negative integral parameter differences

New explicit as well as manifestly symmetric three-term summation formulas are derived for the Clausenian hypergeometric series 3F2(1)3F2(1) with negative integral parameter differences. Our results generalize and naturally extend several similar relations published, in recent years, by many authors. An appropriate and useful connection is established with the quite underestimated 1974 paper by P.W. Karlsson.