If you want Bootstrap 3 support for RTL and LTR on your site, you can modify CSS rules "on the fly", attached here is a function, it modifies the major classes for Bootstrap 3 like col-(xs|sm|md|lg)-(1-12), col-(xs|sm|md|lg)-push-(1-12), col-(xs|sm|md|lg)-pull-(1-12), col-(xs|sm|md|lg)-offset-(1-12), there are many more classes to be modified but i needed only those.
All you need to do is call the function layout.setDirection('rtl') or layout.setDirection('ltr') and it will change the CSS Rules for Bootstrap 3 grid system.
Should work on all browsers (IE>=9).
        var layout = {};
        layout.setDirection = function (direction) {
            layout.rtl = (direction === 'rtl');
            document.getElementsByTagName("html")[0].style.direction = direction;
            var styleSheets = document.styleSheets;
            var modifyRule = function (rule) {
                if (rule.style.getPropertyValue(layout.rtl ? 'left' : 'right') && rule.selectorText.match(/\.col-(xs|sm|md|lg)-push-\d\d*/)) {
                    rule.style.setProperty((layout.rtl ? 'right' : 'left'), rule.style.getPropertyValue((layout.rtl ? 'left' : 'right')));
                    rule.style.removeProperty((layout.rtl ? 'left' : 'right'));
                }
                if (rule.style.getPropertyValue(layout.rtl ? 'right' : 'left') && rule.selectorText.match(/\.col-(xs|sm|md|lg)-pull-\d\d*/)) {
                    rule.style.setProperty((layout.rtl ? 'left' : 'right'), rule.style.getPropertyValue((layout.rtl ? 'right' : 'left')));
                    rule.style.removeProperty((layout.rtl ? 'right' : 'left'));
                }
                if (rule.style.getPropertyValue(layout.rtl ? 'margin-left' : 'margin-right') && rule.selectorText.match(/\.col-(xs|sm|md|lg)-offset-\d\d*/)) {
                    rule.style.setProperty((layout.rtl ? 'margin-right' : 'margin-left'), rule.style.getPropertyValue((layout.rtl ? 'margin-left' : 'margin-right')));
                    rule.style.removeProperty((layout.rtl ? 'margin-left' : 'margin-right'));
                }
                if (rule.style.getPropertyValue('float') && rule.selectorText.match(/\.col-(xs|sm|md|lg)-\d\d*/)) {
                    rule.style.setProperty('float', (layout.rtl ? 'right' : 'left'));
                }
            };
            try {
                for (var i = 0; i < styleSheets.length; i++) {
                    var rules = styleSheets[i].cssRules || styleSheets[i].rules;
                    if (rules) {
                        for (var j = 0; j < rules.length; j++) {
                            if (rules[j].type === 4) {
                                var mediaRules = rules[j].cssRules || rules[j].rules
                                for (var y = 0; y < mediaRules.length; y++) {
                                    modifyRule(mediaRules[y]);
                                }
                            }
                            if (rules[j].type === 1) {
                                modifyRule(rules[j]);
                            }
                        }
                    }
                }
            } catch (e) {
                // Firefox might throw a SecurityError exception but it will work
                if (e.name !== 'SecurityError') {
                    throw e;
                }
            }
        }