In 2000 and 2004, the state voted for President George W. Bush. This November, they could vote for presumptive Republican nominee John McCain.
Mrs Clinton won the state handily on Tuesday, but the Democrats may not be able to retain it if Mr Obama becomes the party nominee.
Exit polls show that only 36 per cent of Clinton supporters would vote for him in the presidential election. About as many said they would cast their votes for Mr McCain instead.
Mr Obama's voters are polarised too. Some 51 per cent of them said they would back Mrs Clinton were she the nominee and 31 per cent said they would vote for Mr McCain.
Two in 10 voters here said race was a factor in their votes. More than eight out of 10 voters who said so backed Mrs Clinton.
States with West Virginia-like mix of voters - Ohio, Pennsylvania, New Hampshire and Kentucky - may also resist the candidacy of Mr Obama if he becomes the first black Democratic nominee.
BHAGYASHREE GAREKAR