The phenomenon you're observing, that of negative polarity items seeming to form another factor, independent of the (intended) affine items with positive polarity, is not uncommon. The concern is whether this represents a genuinely distinct dimension as opposed to merely the other "half" of a bipolar dimension. For this reason, some argue against using negative polarity items (e.g., Pilotte, W. J., & Gable, R. K. (1990). The impact of positive and negative item stems on the validity of a computer anxiety scale. Educational and Psychological Measurement, 50, 603-610. doi:10.1177/0013164490503016)
One way to explore this would be to look at an oblique rotation of the two factors (or however many you happen to have), and see whether an ignorable or non-ignorable, negative relationship between the factors exists. Alternatively, you could pursue this as a second-order factor model (which, obviously, requires correlated factors).
As well, it's been suggested that some respondent carelessness could yield the same type of pattern (https://conservancy.umn.edu/bitstream/handle/11299/102192/1/v09n4p367.pdf).
This often occurs when researchers use complex wording so that they can reverse score more items. These can become confusing items and that messes up the scale. First I'd check if they are more awkwardly phrased. Do you have response latencies for the items? And as David Morse says, you can treat this as another factor, but it is worth thinking why they may be that way.
"Some researchers have attempted to reduce the potential effects of response pattern biases by incorporating negatively worded or reverse-coded items on their questionnaires (cf. Hinkin, 1995; Idaszak & Drasgow, 1987). The basic logic here is that reverse-coded items are like cognitive “speed bumps” that require respondents to engage in more controlled, as opposed to automatic, cognitive processing. Unfortunately, research has shown that reverse-coded items may produce artifactual response factors consisting exclusively of negatively worded items (Harvey, Billings, & Nilan, 1985) that may
disappear after the reverse-coded items are rewritten in a positive manner (Idaszak & Drasgow, 1987). Schmitt and Stults (1986) have argued that the effects of negatively worded items may occur
because once respondents establish a pattern of responding to a questionnaire, they may fail to attend to the positive–negative wording of the items. In addition, they have shown that factors representing negatively worded items may occur in cases where as few as 10% of the respondents fail to recognize that some items are reverse coded. Thus, negatively worded items may be a source of method bias." Refer to Podsakoff et al., 2003.